id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-12700 | Product of Coprime Numbers whose Divisor Sum is Square has Square Divisor Sum | Let $m, n \in \Z_{>0}$ be a positive integer.
Let the divisor sum of $m$ and $n$ both be square.
Let $m$ and $n$ be coprime.
Then the divisor sum of $m n$ is square. | Let $\map {\sigma_1} m = k^2$.
Let $\map {\sigma_1} n = l^2$.
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {m n}
| r = \map {\sigma_1} m \map {\sigma_1} n
| c = Divisor Sum Function is Multiplicative
}}
{{eqn | r = k^2 l^2
| c = from above
}}
{{eqn | r = \paren {k l}^2
| c = from above
}}
{{e... | Let $m, n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
Let the [[Definition:Divisor Sum Function|divisor sum]] of $m$ and $n$ both be [[Definition:Square Number|square]].
Let $m$ and $n$ be [[Definition:Coprime Integers|coprime]].
Then the [[Definition:Divisor Sum Function|divisor sum]] of $m... | Let $\map {\sigma_1} m = k^2$.
Let $\map {\sigma_1} n = l^2$.
Thus:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {m n}
| r = \map {\sigma_1} m \map {\sigma_1} n
| c = [[Divisor Sum Function is Multiplicative]]
}}
{{eqn | r = k^2 l^2
| c = from above
}}
{{eqn | r = \paren {k l}^2
| c = from above
... | Product of Coprime Numbers whose Divisor Sum is Square has Square Divisor Sum | https://proofwiki.org/wiki/Product_of_Coprime_Numbers_whose_Divisor_Sum_is_Square_has_Square_Divisor_Sum | https://proofwiki.org/wiki/Product_of_Coprime_Numbers_whose_Divisor_Sum_is_Square_has_Square_Divisor_Sum | [
"Numbers whose Divisor Sum is Square",
"Coprime Integers"
] | [
"Definition:Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Square Number",
"Definition:Coprime/Integers",
"Definition:Divisor Sum Function",
"Definition:Square Number"
] | [
"Divisor Sum Function is Multiplicative",
"Category:Numbers whose Divisor Sum is Square",
"Category:Coprime Integers"
] |
proofwiki-12701 | Sequence of Differences on Generalized Pentagonal Numbers | Recall the generalised pentagonal numbers $GP_n$ for $n = 0, 1, 2, \ldots$
Consider the sequence defined as $\Delta_n = GP_{n + 1} - GP_n$:
:$1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, \ldots$
{{OEIS|A026741}}
Then:
:The values of $\Delta_n$ for odd $n$ consist of the odd numbers
:The values of $\Delta_n$ for e... | Recall the definition of the generalised pentagonal numbers $GP_n$ for $n = 0, 1, 2, \ldots$
:$GP_n = \begin{cases} \dfrac {m \paren {3 m + 1} } 2 & : n = 2 m \\
\dfrac {m \paren {3 m - 1} } 2 & : n = 2 m - 1 \end{cases}$
for $n = 0, 1, 2, \ldots$
Hence:
{{begin-eqn}}
{{eqn | l = \Delta_{2 n - 1}
| r = GP_{2 n} -... | Recall the [[Definition:Generalized Pentagonal Number|generalised pentagonal numbers]] $GP_n$ for $n = 0, 1, 2, \ldots$
Consider the [[Definition:Integer Sequence|sequence]] defined as $\Delta_n = GP_{n + 1} - GP_n$:
:$1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, \ldots$
{{OEIS|A026741}}
Then:
:The values of $\... | Recall the definition of the [[Definition:Generalized Pentagonal Number/Definition 2|generalised pentagonal numbers]] $GP_n$ for $n = 0, 1, 2, \ldots$
:$GP_n = \begin{cases} \dfrac {m \paren {3 m + 1} } 2 & : n = 2 m \\
\dfrac {m \paren {3 m - 1} } 2 & : n = 2 m - 1 \end{cases}$
for $n = 0, 1, 2, \ldots$
Hence:
{{beg... | Sequence of Differences on Generalized Pentagonal Numbers | https://proofwiki.org/wiki/Sequence_of_Differences_on_Generalized_Pentagonal_Numbers | https://proofwiki.org/wiki/Sequence_of_Differences_on_Generalized_Pentagonal_Numbers | [
"Generalized Pentagonal Numbers"
] | [
"Definition:Generalized Pentagonal Number",
"Definition:Integer Sequence",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Natural Numbers"
] | [
"Definition:Generalized Pentagonal Number/Definition 2",
"Definition:Sequence",
"Definition:Natural Numbers",
"Definition:Sequence",
"Definition:Odd Integer"
] |
proofwiki-12702 | Intersection of Semilattice Ideals is Ideal/Set of Sets | Let $\struct {S, \preceq}$ be a bounded below join semilattice.
Let $\II$ be a set of ideals in $\struct {S, \preceq}$.
Then $\bigcap \II$ is an ideal in $\struct {S, \preceq}$. | === Non-Empty Set ===
By Bottom in Ideal:
:$\forall I \in \II: \bot \in I$
where $\bot$ denotes the smallest element in $S$.
By definition of intersection:
:$\bot \in \bigcap \II$
Hence $\bigcap \II$ is non-empty.
{{qed|lemma}} | Let $\struct {S, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $\II$ be a [[Definition:Set of Sets|set]] of [[Definition:Ideal (Order Theory)|ideals]] in $\struct {S, \preceq}$.
Then $\bigcap \II$ is an [[Definition:Ideal (Order Theory)|ideal]] in... | === Non-Empty Set ===
By [[Bottom in Ideal]]:
:$\forall I \in \II: \bot \in I$
where $\bot$ denotes the [[Definition:Smallest Element|smallest element]] in $S$.
By definition of [[Definition:Intersection of Set of Sets|intersection]]:
:$\bot \in \bigcap \II$
Hence $\bigcap \II$ is [[Definition:Non-Empty Set|non-empt... | Intersection of Semilattice Ideals is Ideal/Set of Sets | https://proofwiki.org/wiki/Intersection_of_Semilattice_Ideals_is_Ideal/Set_of_Sets | https://proofwiki.org/wiki/Intersection_of_Semilattice_Ideals_is_Ideal/Set_of_Sets | [
"Set Intersection",
"Join and Meet Semilattices"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Definition:Ideal (Order Theory)"
] | [
"Bottom in Ideal",
"Definition:Smallest Element",
"Definition:Set Intersection/Set of Sets",
"Definition:Non-Empty Set",
"Definition:Set Intersection/Set of Sets",
"Definition:Set Intersection/Set of Sets",
"Definition:Set Intersection/Set of Sets",
"Definition:Set Intersection/Set of Sets"
] |
proofwiki-12703 | Compact Element iff Existence of Finite Subset that Element equals Intersection and Includes Subset | Let $X, E$ be sets.
Let $P = \struct {\powerset X, \precsim}$ be an inclusion ordered set
where
:$\powerset X$ denotes the power set of $X$
:$\mathord \precsim = \mathord \subseteq \cap \paren {\powerset X \times \powerset X}$
Let $L = \struct {S, \preceq}$ be a continuous lattice subframe of $P$.
Then $E$ is compact e... | By Power Set is Complete Lattice:
:$P$ is a complete lattice.
By Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice:
:$L$ is a complete lattice.
By Image of Operator Generated by Closure System is Set of Closure System:
:$\map {\operatorname {operator} } L \sqbrk {\powerset X} = S$
where $\map {\o... | Let $X, E$ be [[Definition:Set|sets]].
Let $P = \struct {\powerset X, \precsim}$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]]
where
:$\powerset X$ denotes the [[Definition:Power Set|power set]] of $X$
:$\mathord \precsim = \mathord \subseteq \cap \paren {\powerset X \times \powerset X}... | By [[Power Set is Complete Lattice]]:
:$P$ is a [[Definition:Complete Lattice|complete lattice]].
By [[Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice]]:
:$L$ is a [[Definition:Complete Lattice|complete lattice]].
By [[Image of Operator Generated by Closure System is Set of Closure System]]:
... | Compact Element iff Existence of Finite Subset that Element equals Intersection and Includes Subset | https://proofwiki.org/wiki/Compact_Element_iff_Existence_of_Finite_Subset_that_Element_equals_Intersection_and_Includes_Subset | https://proofwiki.org/wiki/Compact_Element_iff_Existence_of_Finite_Subset_that_Element_equals_Intersection_and_Includes_Subset | [
"Join and Meet Semilattices",
"Way Below Relation"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Power Set",
"Definition:Continuous Lattice Subframe",
"Definition:Compact Element",
"Definition:Set of Sets",
"Definition:Finite Subset"
] | [
"Power Set is Complete Lattice",
"Definition:Complete Lattice",
"Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice",
"Definition:Complete Lattice",
"Image of Operator Generated by Closure System is Set of Closure System",
"Definition:Operator Generated by Ordered Subset",
"Closure... |
proofwiki-12704 | Legendre's Condition | Let $y =\map y x$ be a real function, such that:
:$\map y a = A,\quad \map y b = B$
Let $J \sqbrk y$ be a functional, such that:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
where
:$F \in C^2 \closedint a b$
{{WRT}} all its variables, and $C$ stands for differentiability class.
Then a necessary condition for $... | === Lemma 1 ===
Let $y = \map y x$ be a real function, such that:
:$\map y a = A, \quad \map y b = B$
Let $J \sqbrk y$ be a functional, such that:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
where:
:$F \in C^2 \closedint a b$
{{WRT}} all its variables.
Then:
:$\ds \delta^2 J \sqbrk {y; h} = \int_a^b \paren {\... | Let $y =\map y x$ be a [[Definition:Real Function|real function]], such that:
:$\map y a = A,\quad \map y b = B$
Let $J \sqbrk y$ be a [[Definition:Real Functional|functional]], such that:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
where
:$F \in C^2 \closedint a b$
{{WRT}} all its [[Definition:Independ... | === [[Legendre's Condition/Lemma 1|Lemma 1]] ===
Let $y = \map y x$ be a [[Definition:Real Function|real function]], such that:
:$\map y a = A, \quad \map y b = B$
Let $J \sqbrk y$ be a [[Definition:Real Functional|functional]], such that:
:$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$
where:
:$F \in C^2 \c... | Legendre's Condition | https://proofwiki.org/wiki/Legendre's_Condition | https://proofwiki.org/wiki/Legendre's_Condition | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Functional/Real",
"Definition:Independent Variable",
"Definition:Differentiability Class",
"Definition:Conditional/Necessary Condition",
"Definition:Minimum Value of Functional"
] | [
"Legendre's Condition/Lemma 1",
"Definition:Real Function",
"Definition:Functional/Real",
"Definition:Independent Variable",
"Definition:Real Function"
] |
proofwiki-12705 | Euler's Pentagonal Numbers Theorem/Corollary 1 | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $\map {\sigma_1} n$ denote the divisor sum of $n$.
Then:
:$\map {\sigma_1} n = \ds \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map {\sigma_1} {n - GP_k} + n \sqbrk {\exists k \in \Z: GP_k = n}$ | {{ProofWanted|Follows somehow from Euler's Pentagonal Numbers Theorem, but at this time of night I have not a clue how.}} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\map {\sigma_1} n$ denote the [[Definition:Divisor Sum Function|divisor sum]] of $n$.
Then:
:$\map {\sigma_1} n = \ds \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map {\sigma_1} {n - GP_k}... | {{ProofWanted|Follows somehow from [[Euler's Pentagonal Numbers Theorem]], but at this time of night I have not a clue how.}} | Euler's Pentagonal Numbers Theorem/Corollary 1 | https://proofwiki.org/wiki/Euler's_Pentagonal_Numbers_Theorem/Corollary_1 | https://proofwiki.org/wiki/Euler's_Pentagonal_Numbers_Theorem/Corollary_1 | [
"Generalized Pentagonal Numbers",
"Divisor Sum Function",
"Euler's Pentagonal Numbers Theorem"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Divisor Sum Function"
] | [
"Euler's Pentagonal Numbers Theorem"
] |
proofwiki-12706 | Euler's Pentagonal Numbers Theorem/Corollary 2 | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $\map p n$ denote the number of partitions on $n$.
Then:
:$\map p n = \ds \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map p {n - GP_k} + \sqbrk {\exists k \in \Z: GP_k = n}$ | {{ProofWanted|Follows somehow from Euler's Pentagonal Numbers Theorem, but at this time of night I have not a clue how.}} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\map p n$ denote the number of [[Definition:Integer Partition|partitions]] on $n$.
Then:
:$\map p n = \ds \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map p {n - GP_k} + \sqbrk {\exists k ... | {{ProofWanted|Follows somehow from [[Euler's Pentagonal Numbers Theorem]], but at this time of night I have not a clue how.}} | Euler's Pentagonal Numbers Theorem/Corollary 2 | https://proofwiki.org/wiki/Euler's_Pentagonal_Numbers_Theorem/Corollary_2 | https://proofwiki.org/wiki/Euler's_Pentagonal_Numbers_Theorem/Corollary_2 | [
"Generalized Pentagonal Numbers",
"Partition Theory",
"Euler's Pentagonal Numbers Theorem"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Integer Partition"
] | [
"Euler's Pentagonal Numbers Theorem"
] |
proofwiki-12707 | Long Period Prime/Examples/23 | The prime number $23$ is a long period prime:
:$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ | From Reciprocal of $23$:
{{:Reciprocal of 23}}
Counting the digits, it is seen that this has a period of recurrence of $22$.
Hence the result.
{{qed}} | The [[Definition:Prime Number|prime number]] $23$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ | From [[Reciprocal of 23|Reciprocal of $23$]]:
{{:Reciprocal of 23}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $22$.
Hence the result.
{{qed}} | Long Period Prime/Examples/23 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/23 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/23 | [
"23",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime"
] | [
"Reciprocal of 23",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-12708 | 23 is Largest Integer not Sum of Distinct Perfect Powers | The largest integer which cannot be expressed as the sum of distinct perfect powers is $23$. | The first few perfect powers are:
:$1, 4, 8, 9, 16, 25, 27, 32, \dots$
First we show that $23$ cannot be expressed as the sum of distinct perfect powers.
Only $1, 4, 8, 9, 16$ are perfect powers less than $23$.
Suppose $23$ can be so expressed.
Since $1 + 4 + 8 + 9 = 22 < 23$, $16$ must be used in the sum.
However $23 ... | The largest [[Definition:Integer|integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Perfect Power|perfect powers]] is $23$. | The first few [[Definition:Perfect Power|perfect powers]] are:
:$1, 4, 8, 9, 16, 25, 27, 32, \dots$
First we show that $23$ cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Perfect Power|perfect powers]].
Only $1, 4, 8, 9, 16$ are [[Definition:Perfect Po... | 23 is Largest Integer not Sum of Distinct Perfect Powers | https://proofwiki.org/wiki/23_is_Largest_Integer_not_Sum_of_Distinct_Perfect_Powers | https://proofwiki.org/wiki/23_is_Largest_Integer_not_Sum_of_Distinct_Perfect_Powers | [
"23",
"Powers"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Perfect Power"
] | [
"Definition:Perfect Power",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Perfect Power",
"Definition:Perfect Power",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Perfect Power",
"Richert's... |
proofwiki-12709 | Smallest Integer not Sum of Two Ulam Numbers | The smallest integer greater than $1$ which is not the sum of two Ulam numbers is $23$. | Recall the Ulam numbers:
{{:Definition:Ulam Number/Sequence}}
We have:
{{begin-eqn}}
{{eqn | l = 2
| r = 1 + 1
}}
{{eqn | l = 3
| r = 2 + 1
}}
{{eqn | l = 4
| r = 3 + 1
}}
{{eqn | r = 2 + 2
}}
{{eqn | l = 5
| r = 4 + 1
}}
{{eqn | r = 3 + 2
}}
{{eqn | l = 6
| r = 4 + 2
}}
{{eqn | r = 3 + 3
... | The smallest [[Definition:Integer|integer]] greater than $1$ which is not the [[Definition:Integer Addition|sum]] of two [[Definition:Ulam Number|Ulam numbers]] is $23$. | Recall the [[Definition:Ulam Number/Sequence|Ulam numbers]]:
{{:Definition:Ulam Number/Sequence}}
We have:
{{begin-eqn}}
{{eqn | l = 2
| r = 1 + 1
}}
{{eqn | l = 3
| r = 2 + 1
}}
{{eqn | l = 4
| r = 3 + 1
}}
{{eqn | r = 2 + 2
}}
{{eqn | l = 5
| r = 4 + 1
}}
{{eqn | r = 3 + 2
}}
{{eqn | l = 6
... | Smallest Integer not Sum of Two Ulam Numbers | https://proofwiki.org/wiki/Smallest_Integer_not_Sum_of_Two_Ulam_Numbers | https://proofwiki.org/wiki/Smallest_Integer_not_Sum_of_Two_Ulam_Numbers | [
"Ulam Numbers",
"23"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Ulam Number"
] | [
"Definition:Ulam Number/Sequence",
"Definition:Ulam Number",
"Definition:Ulam Number",
"Definition:Ulam Number",
"Definition:Ulam Number",
"Definition:Ulam Number"
] |
proofwiki-12710 | Numbers with Square-Free Binomial Coefficients | For every $n$ greater than $23$, there exists a binomial coefficient $\dbinom n k$ that is not square-free.
More specifically, the list of numbers $n$ such that $\dbinom n k$ are square-free for all $k = 0, \dots, n$ is given by:
:$1, 2, 3, 5, 7, 11, 23$
{{OEIS|A048278}} | === Lemma ===
{{:Numbers with Square-Free Binomial Coefficients/Lemma}}{{qed|lemma}} | For every $n$ greater than $23$, there exists a [[Definition:Binomial Coefficient|binomial coefficient]] $\dbinom n k$ that is not [[Definition:Square-Free Integer|square-free]].
More specifically, the list of numbers $n$ such that $\dbinom n k$ are [[Definition:Square-Free Integer|square-free]] for all $k = 0, \dots... | === [[Numbers with Square-Free Binomial Coefficients/Lemma|Lemma]] ===
{{:Numbers with Square-Free Binomial Coefficients/Lemma}}{{qed|lemma}} | Numbers with Square-Free Binomial Coefficients | https://proofwiki.org/wiki/Numbers_with_Square-Free_Binomial_Coefficients | https://proofwiki.org/wiki/Numbers_with_Square-Free_Binomial_Coefficients | [
"Square-Free Integers",
"Binomial Coefficients",
"23"
] | [
"Definition:Binomial Coefficient",
"Definition:Square-Free Integer",
"Definition:Square-Free Integer"
] | [
"Numbers with Square-Free Binomial Coefficients/Lemma",
"Numbers with Square-Free Binomial Coefficients/Lemma",
"Numbers with Square-Free Binomial Coefficients/Lemma",
"Numbers with Square-Free Binomial Coefficients/Lemma"
] |
proofwiki-12711 | Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice | Let $L = \struct {X, \preceq}$ be a complete lattice.
Let $S = \struct {T, \precsim}$ be an infima inheriting ordered subset of $L$.
Then $S$ is a complete lattice. | Let $A$ be subset of $T$.
By definition of complete lattice:
:$A$ admits an infimum in $L$.
Thus by definition of infima inheriting:
:$A$ admits an infimum in $S$.
Hence by dual of Lattice is Complete iff it Admits All Suprema:
:$S$ is a complete lattice.
{{qed}} | Let $L = \struct {X, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $S = \struct {T, \precsim}$ be an [[Definition:Infima Inheriting|infima inheriting]] [[Definition:Ordered Subset|ordered subset]] of $L$.
Then $S$ is a [[Definition:Complete Lattice|complete lattice]]. | Let $A$ be [[Definition:Subset|subset]] of $T$.
By definition of [[Definition:Complete Lattice|complete lattice]]:
:$A$ admits an [[Definition:Infimum of Set|infimum]] in $L$.
Thus by definition of [[Definition:Infima Inheriting|infima inheriting]]:
:$A$ admits an [[Definition:Infimum of Set|infimum]] in $S$.
Hence ... | Infima Inheriting Ordered Subset of Complete Lattice is Complete Lattice | https://proofwiki.org/wiki/Infima_Inheriting_Ordered_Subset_of_Complete_Lattice_is_Complete_Lattice | https://proofwiki.org/wiki/Infima_Inheriting_Ordered_Subset_of_Complete_Lattice_is_Complete_Lattice | [
"Complete Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Infima Inheriting",
"Definition:Ordered Subset",
"Definition:Complete Lattice"
] | [
"Definition:Subset",
"Definition:Complete Lattice",
"Definition:Infimum of Set",
"Definition:Infima Inheriting",
"Definition:Infimum of Set",
"Lattice is Complete iff it Admits All Suprema",
"Definition:Complete Lattice"
] |
proofwiki-12712 | Sum of Reciprocals in Base 10 with Zeroes Removed | The infinite series
:$\ds \sum_{\map P n} \frac 1 n$
where $\map P n$ is the propositional function:
:$\forall n \in \Z_{>0}: \map P n \iff$ the decimal representation of $n$ contains no instances of the digit $0$
converges to the approximate limit $23 \cdotp 10345 \ldots$ | For each $k \in \N$, there are $9^k$ $k$-digit numbers containing no instances of the digit $0$.
Each of these numbers is at least $10^{k - 1}$.
Hence the reciprocals of each of these numbers is at most $\dfrac 1 {10^{k - 1}}$.
Thus:
{{begin-eqn}}
{{eqn | l = \sum_{\map P n} \frac 1 n
| o = <
| r = \sum_{k ... | The [[Definition:Infinite Series|infinite series]]
:$\ds \sum_{\map P n} \frac 1 n$
where $\map P n$ is the [[Definition:Propositional Function|propositional function]]:
:$\forall n \in \Z_{>0}: \map P n \iff$ the [[Definition:Decimal Expansion|decimal representation]] of $n$ contains no instances of the [[Definition:... | For each $k \in \N$, there are $9^k$ $k$-[[Definition:Digit|digit]] numbers containing no instances of the [[Definition:Digit|digit]] $0$.
Each of these numbers is at least $10^{k - 1}$.
Hence the [[Definition:Reciprocal|reciprocals]] of each of these numbers is at most $\dfrac 1 {10^{k - 1}}$.
Thus:
{{begin-eqn}}
{... | Sum of Reciprocals in Base 10 with Zeroes Removed | https://proofwiki.org/wiki/Sum_of_Reciprocals_in_Base_10_with_Zeroes_Removed | https://proofwiki.org/wiki/Sum_of_Reciprocals_in_Base_10_with_Zeroes_Removed | [
"Reciprocals",
"Series"
] | [
"Definition:Series",
"Definition:Propositional Function",
"Definition:Decimal Expansion",
"Definition:Digit",
"Definition:Convergent Series",
"Definition:Limit of Sequence/Real Numbers"
] | [
"Definition:Digit",
"Definition:Digit",
"Definition:Reciprocal",
"Sum of Geometric Sequence",
"Definition:Convergent Series",
"Closed Form for Triangular Numbers/Direct Proof"
] |
proofwiki-12713 | Gelfond's Constant minus Pi | Gelfond's constant minus $\pi$ is very close to $20$:
:$e^\pi - \pi \approx 20$ | We have:
:$e^\pi \approx 23 \cdotp 14069 \, 26327 \, 79269 \ldots$
:$\pi \approx 3 \cdotp 14159 \, 26535 \, 89793$
Then:
<pre>
23.14069 26327 79269
- 3.14159 26535 89793
---------------------
19.99909 99791 89486
---------------------
</pre>{{qed}} | [[Definition:Gelfond's Constant|Gelfond's constant]] minus $\pi$ is very close to $20$:
:$e^\pi - \pi \approx 20$ | We have:
:$e^\pi \approx 23 \cdotp 14069 \, 26327 \, 79269 \ldots$
:$\pi \approx 3 \cdotp 14159 \, 26535 \, 89793$
Then:
<pre>
23.14069 26327 79269
- 3.14159 26535 89793
---------------------
19.99909 99791 89486
---------------------
</pre>{{qed}} | Gelfond's Constant minus Pi | https://proofwiki.org/wiki/Gelfond's_Constant_minus_Pi | https://proofwiki.org/wiki/Gelfond's_Constant_minus_Pi | [
"Pi",
"Gelfond's Constant",
"Approximate Relations between Pi and Euler's Number"
] | [
"Definition:Gelfond's Constant"
] | [] |
proofwiki-12714 | Sums of Consecutive Sequences of Squares that equal Squares | Apart from $1$, the $24$th square pyramidal number is the only one which is square:
:$1^2 + 2^2 + 3^2 + \cdots + 24^2 = 70^2$
while there are several Sum of Sequence of Squares which are square, for example:
:$18^2 + 19^2 + \cdots + 28^2 = 77^2$
and:
:$25^2 + 26^2 + \cdots + 624^2 = 9010^2$ | We have:
{{begin-eqn}}
{{eqn | l = 1^2 + 2^2 + 3^2 + \cdots + 24^2
| r = \dfrac {24 \times \paren {24 + 1} \times \paren {2 \times 24 + 1} } 6
| c = Sum of Sequence of Squares
}}
{{eqn | r = \dfrac {24 \times 25 \times 49} 6
| c =
}}
{{eqn | r = \dfrac {2^3 \times 3 \times 5^2 \times 7^2} {2 \times 3... | Apart from $1$, the $24$th [[Definition:Square Pyramidal Number|square pyramidal number]] is the only one which is [[Definition:Square Number|square]]:
:$1^2 + 2^2 + 3^2 + \cdots + 24^2 = 70^2$
while there are several [[Sum of Sequence of Squares]] which are [[Definition:Square Number|square]], for example:
:$18^2 + 1... | We have:
{{begin-eqn}}
{{eqn | l = 1^2 + 2^2 + 3^2 + \cdots + 24^2
| r = \dfrac {24 \times \paren {24 + 1} \times \paren {2 \times 24 + 1} } 6
| c = [[Sum of Sequence of Squares]]
}}
{{eqn | r = \dfrac {24 \times 25 \times 49} 6
| c =
}}
{{eqn | r = \dfrac {2^3 \times 3 \times 5^2 \times 7^2} {2 \ti... | Sums of Consecutive Sequences of Squares that equal Squares | https://proofwiki.org/wiki/Sums_of_Consecutive_Sequences_of_Squares_that_equal_Squares | https://proofwiki.org/wiki/Sums_of_Consecutive_Sequences_of_Squares_that_equal_Squares | [
"Square Numbers",
"Sums of Sequences",
"Sum of Sequence of Squares"
] | [
"Definition:Square Pyramidal Number",
"Definition:Square Number",
"Sum of Sequence of Squares",
"Definition:Square Number"
] | [
"Sum of Sequence of Squares",
"Sum of Sequence of Squares",
"Sum of Sequence of Squares"
] |
proofwiki-12715 | Sum of Squares of Divisors of 24 and 26 are Equal | The sum of the squares of the divisors of $24$ equals the sum of the squares of the divisors of $26$:
:$\map {\sigma_2} {24} = \map {\sigma_2} {26}$
where $\sigma_\alpha$ denotes the divisor function. | The divisors of $24$ are:
:$1, 2, 3, 4, 6, 8, 12, 24$
The divisors of $26$ are:
:$1, 2, 13, 26$
Then we have:
{{begin-eqn}}
{{eqn | r = 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 8^2 + 12^2 + 24^2
| o =
| c =
}}
{{eqn | r = 1 + 4 + 9 + 16 + 36 + 64 + 144 + 576
| c =
}}
{{eqn | r = 850
| c =
}}
{{end-eqn}... | The [[Definition:Integer Addition|sum]] of the [[Definition:Square (Algebra)|squares]] of the [[Definition:Divisor of Integer|divisors]] of $24$ equals the [[Definition:Integer Addition|sum]] of the [[Definition:Square (Algebra)|squares]] of the [[Definition:Divisor of Integer|divisors]] of $26$:
:$\map {\sigma_2} {24... | The [[Definition:Divisor of Integer|divisors]] of $24$ are:
:$1, 2, 3, 4, 6, 8, 12, 24$
The [[Definition:Divisor of Integer|divisors]] of $26$ are:
:$1, 2, 13, 26$
Then we have:
{{begin-eqn}}
{{eqn | r = 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 8^2 + 12^2 + 24^2
| o =
| c =
}}
{{eqn | r = 1 + 4 + 9 + 16 + 36 + 6... | Sum of Squares of Divisors of 24 and 26 are Equal | https://proofwiki.org/wiki/Sum_of_Squares_of_Divisors_of_24_and_26_are_Equal | https://proofwiki.org/wiki/Sum_of_Squares_of_Divisors_of_24_and_26_are_Equal | [
"Divisors",
"Square Numbers",
"24",
"26"
] | [
"Definition:Addition/Integers",
"Definition:Square/Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Square/Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Function"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12716 | Smallest Scalene Obtuse Triangle with Integer Sides and Area | The smallest scalene obtuse triangle with integer sides and area has sides of length $4, 13, 15$. | From Heron's Formula, the area $A$ of $\triangle ABC$ is given by:
:$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
Here we have:
{{begin-eqn}}
{{eqn | l = s
| r = \dfrac {4 + 13 + 15} 2
| c =
}}
{{eqn | r = 16
| ... | The smallest [[Definition:Scalene Triangle|scalene]] [[Definition:Obtuse Triangle|obtuse triangle]] with [[Definition:Integer|integer]] [[Definition:Side of Polygon|sides]] and [[Definition:Area|area]] has [[Definition:Side of Polygon|sides]] of [[Definition:Length of Line|length]] $4, 13, 15$. | From [[Heron's Formula]], the [[Definition:Area|area]] $A$ of $\triangle ABC$ is given by:
:$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where $s = \dfrac {a + b + c} 2$ is the [[Definition:Semiperimeter|semiperimeter]] of $\triangle ABC$.
Here we have:
{{begin-eqn}}
{{eqn | l = s
| r = \dfrac ... | Smallest Scalene Obtuse Triangle with Integer Sides and Area | https://proofwiki.org/wiki/Smallest_Scalene_Obtuse_Triangle_with_Integer_Sides_and_Area | https://proofwiki.org/wiki/Smallest_Scalene_Obtuse_Triangle_with_Integer_Sides_and_Area | [
"Obtuse Triangles",
"Scalene Triangles"
] | [
"Definition:Triangle (Geometry)/Scalene",
"Definition:Triangle (Geometry)/Obtuse",
"Definition:Integer",
"Definition:Polygon/Side",
"Definition:Area",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length"
] | [
"Heron's Formula",
"Definition:Area",
"Definition:Semiperimeter"
] |
proofwiki-12717 | Image of Compact Subset under Directed Suprema Preserving Closure Operator | Let $L = \struct {S, \preceq}$ be a bounded below algebric lattice.
Let $c: S \to S$ be a closure operator that preserves directed suprema.
Then:
:$c \sqbrk {\map K L} = \map K {\struct {c \sqbrk S, \precsim} }$
where
:$\map K L$ denotes the compact subset of $L$,
:$c \sqbrk S$ denotes the image of $S$ under $c$,
:$\ma... | We will prove that:
:$\map K {\struct {c \sqbrk S, \precsim} } \subseteq c \sqbrk {\map K L}$
By Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset:
:$c \sqbrk {\map K L} \subseteq \map K {\struct {c \sqbrk S, \precsim} }$
Thus the result by definition of set equality... | Let $L = \struct {S, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebric]] [[Definition:Lattice (Order Theory)|lattice]].
Let $c: S \to S$ be a [[Definition:Closure Operator|closure operator]] that [[Definition:Mapping Preserves Supremum/Directed|preserves directed... | We will prove that:
:$\map K {\struct {c \sqbrk S, \precsim} } \subseteq c \sqbrk {\map K L}$
By [[Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset]]:
:$c \sqbrk {\map K L} \subseteq \map K {\struct {c \sqbrk S, \precsim} }$
Thus the result by definition of [[Defi... | Image of Compact Subset under Directed Suprema Preserving Closure Operator | https://proofwiki.org/wiki/Image_of_Compact_Subset_under_Directed_Suprema_Preserving_Closure_Operator | https://proofwiki.org/wiki/Image_of_Compact_Subset_under_Directed_Suprema_Preserving_Closure_Operator | [
"Continuous Lattices",
"Way Below Relation"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Closure Operator",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Compact Subset of Lattice",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset",
"Definition:Set Equality"
] |
proofwiki-12718 | Smallest Positive Integer with 5 Fibonacci Partitions | The smallest positive integer which can be partitioned into distinct Fibonacci numbers in $5$ different ways is $24$. | {{begin-eqn}}
{{eqn | l = 1
| r = 1
| c = $1$ way
}}
{{eqn | l = 2
| r = 2
| c = $1$ way
}}
{{eqn | l = 3
| r = 3
}}
{{eqn | r = 2 + 1
| c = $2$ ways
}}
{{eqn | l = 4
| r = 3 + 1
| c = $1$ way
}}
{{eqn | l = 5
| r = 5
}}
{{eqn | r = 3 + 2
| c = $2$ ways
}}
{{e... | The smallest [[Definition:Positive Integer|positive integer]] which can be [[Definition:Integer Partition|partitioned]] into [[Definition:Distinct|distinct]] [[Definition:Fibonacci Numbers|Fibonacci numbers]] in $5$ different ways is $24$. | {{begin-eqn}}
{{eqn | l = 1
| r = 1
| c = $1$ way
}}
{{eqn | l = 2
| r = 2
| c = $1$ way
}}
{{eqn | l = 3
| r = 3
}}
{{eqn | r = 2 + 1
| c = $2$ ways
}}
{{eqn | l = 4
| r = 3 + 1
| c = $1$ way
}}
{{eqn | l = 5
| r = 5
}}
{{eqn | r = 3 + 2
| c = $2$ ways
}}
{{e... | Smallest Positive Integer with 5 Fibonacci Partitions | https://proofwiki.org/wiki/Smallest_Positive_Integer_with_5_Fibonacci_Partitions | https://proofwiki.org/wiki/Smallest_Positive_Integer_with_5_Fibonacci_Partitions | [
"Fibonacci Numbers",
"24"
] | [
"Definition:Positive/Integer",
"Definition:Integer Partition",
"Definition:Distinct",
"Definition:Fibonacci Number"
] | [] |
proofwiki-12719 | Numbers Divisible by Sum and Product of Digits | The sequence of positive integers which are divisible by both the sum and product of its digits begins:
:$1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 111, 112, 132, 135, \ldots$
{{OEIS|A038186}} | Let $S$ be the set of all positive integers which are divisible by both the sum and product of their digits.
Trivially, the sum and product of the digits of a one-digit number $n$ are themselves $n$.
Thus from Integer Divides Itself, the positive integers from $1$ to $9$ are in $S$.
The product of any integer with a $0... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] which are [[Definition:Divisor of Integer|divisible]] by both the [[Definition:Integer Addition|sum]] and [[Definition:Integer Multiplication|product]] of its [[Definition:Digit|digits]] begins:
:$1, 2, 3, 4, 5, 6, 7, 8, ... | Let $S$ be the [[Definition:Set|set]] of all [[Definition:Positive Integer|positive integers]] which are [[Definition:Divisor of Integer|divisible]] by both the [[Definition:Integer Addition|sum]] and [[Definition:Integer Multiplication|product]] of their [[Definition:Digit|digits]].
Trivially, the [[Definition:Intege... | Numbers Divisible by Sum and Product of Digits | https://proofwiki.org/wiki/Numbers_Divisible_by_Sum_and_Product_of_Digits | https://proofwiki.org/wiki/Numbers_Divisible_by_Sum_and_Product_of_Digits | [
"Number Theory"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Multiplication/Integers",
"Definition:Digit"
] | [
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Integers",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Digit",
"Intege... |
proofwiki-12720 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$115 = 5 \times 23$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {1... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | From [[Divisor Sum of Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$115 = 5 \times 23$
Hence:
{{... | Divisor Sum of Non-Square Semiprime/Examples/115/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/115/Proof_1 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition",
"Difference of Two Squares"
] |
proofwiki-12721 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | We have that:
:$115 = 5 \times 23$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {115}
| r = \paren {5 + 1} \paren {23 + 1}
| c = Divisor Sum of Non-Square Semiprime
}}
{{eqn | r = 6 \times 24
| c =
}}
{{eqn | r = \paren {2 \tim... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | We have that:
:$115 = 5 \times 23$
and so by definition is a [[Definition:Semiprime Number|semiprime]] whose [[Definition:Prime Factor|prime factors]] are [[Definition:Distinct|distinct]].
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {115}
| r = \paren {5 + 1} \paren {23 + 1}
| c = [[Divisor Sum of No... | Divisor Sum of Non-Square Semiprime/Examples/115/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/115/Proof_2 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Semiprime Number",
"Definition:Prime Factor",
"Definition:Distinct",
"Divisor Sum of Non-Square Semiprime"
] |
proofwiki-12722 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$14 = 2 \times 7$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {14}... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | From [[Divisor Sum of Integer]]
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$14 = 2 \times 7$
Hence:
{{beg... | Divisor Sum of Non-Square Semiprime/Examples/14/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/14/Proof_1 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition"
] |
proofwiki-12723 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | We have that:
:$14 = 2 \times 7$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {14}
| r = \paren {2 + 1} \paren {7 + 1}
| c = Divisor Sum of Non-Square Semiprime
}}
{{eqn | r = 3 \times 8
| c =
}}
{{eqn | r = 24
| c =
}}
... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | We have that:
:$14 = 2 \times 7$
and so by definition is a [[Definition:Semiprime Number|semiprime]] whose [[Definition:Prime Factor|prime factors]] are [[Definition:Distinct|distinct]].
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {14}
| r = \paren {2 + 1} \paren {7 + 1}
| c = [[Divisor Sum of Non-Sq... | Divisor Sum of Non-Square Semiprime/Examples/14/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/14/Proof_2 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Semiprime Number",
"Definition:Prime Factor",
"Definition:Distinct",
"Divisor Sum of Non-Square Semiprime"
] |
proofwiki-12724 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$15 = 3 \times 5$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {15}
... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | From [[Divisor Sum of Integer]]
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$15 = 3 \times 5$
Hence:
{{beg... | Divisor Sum of Non-Square Semiprime/Examples/15/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/15/Proof_1 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition"
] |
proofwiki-12725 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | We have that:
:$15 = 3 \times 5$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {15}
| r = \paren {3 + 1} \paren {5 + 1}
| c = Divisor Sum of Non-Square Semiprime
}}
{{eqn | r = 4 \times 6
| c =
}}
{{eqn | r = 24
| c =
}}
... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | We have that:
:$15 = 3 \times 5$
and so by definition is a [[Definition:Semiprime Number|semiprime]] whose [[Definition:Prime Factor|prime factors]] are [[Definition:Distinct|distinct]].
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {15}
| r = \paren {3 + 1} \paren {5 + 1}
| c = [[Divisor Sum of Non-Sq... | Divisor Sum of Non-Square Semiprime/Examples/15/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/15/Proof_2 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Semiprime Number",
"Definition:Prime Factor",
"Definition:Distinct",
"Divisor Sum of Non-Square Semiprime"
] |
proofwiki-12726 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$206 = 2 \times 103$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {2... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | From [[Divisor Sum of Integer]]
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$206 = 2 \times 103$
Hence:
{{... | Divisor Sum of Non-Square Semiprime/Examples/206/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/206/Proof_1 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition"
] |
proofwiki-12727 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | We have that:
:$206 = 2 \times 103$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {206}
| r = \paren {2 + 1} \paren {103 + 1}
| c = Divisor Sum of Non-Square Semiprime
}}
{{eqn | r = 3 \times 104
| c =
}}
{{eqn | r = 312
|... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | We have that:
:$206 = 2 \times 103$
and so by definition is a [[Definition:Semiprime Number|semiprime]] whose [[Definition:Prime Factor|prime factors]] are [[Definition:Distinct|distinct]].
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {206}
| r = \paren {2 + 1} \paren {103 + 1}
| c = [[Divisor Sum of ... | Divisor Sum of Non-Square Semiprime/Examples/206/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/206/Proof_2 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Semiprime Number",
"Definition:Prime Factor",
"Definition:Distinct",
"Divisor Sum of Non-Square Semiprime"
] |
proofwiki-12728 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$22 = 2 \times 11$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {22... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | From [[Divisor Sum of Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$22 = 2 \times 11$
Hence:
{{b... | Divisor Sum of Non-Square Semiprime/Examples/22/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/22/Proof_1 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition"
] |
proofwiki-12729 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | We have that:
:$22 = 2 \times 11$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {22}
| r = \paren {2 + 1} \paren {11 + 1}
| c = Divisor Sum of Non-Square Semiprime
}}
{{eqn | r = 3 \times 12
| c =
}}
{{eqn | r = 3 \times \paren ... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | We have that:
:$22 = 2 \times 11$
and so by definition is a [[Definition:Semiprime Number|semiprime]] whose [[Definition:Prime Factor|prime factors]] are [[Definition:Distinct|distinct]].
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {22}
| r = \paren {2 + 1} \paren {11 + 1}
| c = [[Divisor Sum of Non-... | Divisor Sum of Non-Square Semiprime/Examples/22/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/22/Proof_2 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Semiprime Number",
"Definition:Prime Factor",
"Definition:Distinct",
"Divisor Sum of Non-Square Semiprime"
] |
proofwiki-12730 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$26 = 2 \times 13$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {26... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | From [[Divisor Sum of Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$26 = 2 \times 13$
Hence:
{{b... | Divisor Sum of Non-Square Semiprime/Examples/26/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/26/Proof_1 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition"
] |
proofwiki-12731 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | We have that:
:$26 = 2 \times 13$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {26}
| r = \paren {2 + 1} \paren {13 + 1}
| c = Divisor Sum of Non-Square Semiprime
}}
{{eqn | r = 3 \times 14
| c =
}}
{{eqn | r = 42
| c =
... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | We have that:
:$26 = 2 \times 13$
and so by definition is a [[Definition:Semiprime Number|semiprime]] whose [[Definition:Prime Factor|prime factors]] are [[Definition:Distinct|distinct]].
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {26}
| r = \paren {2 + 1} \paren {13 + 1}
| c = [[Divisor Sum of Non-... | Divisor Sum of Non-Square Semiprime/Examples/26/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/26/Proof_2 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Semiprime Number",
"Definition:Prime Factor",
"Definition:Distinct",
"Divisor Sum of Non-Square Semiprime"
] |
proofwiki-12732 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$94 = 2 \times 47$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {94... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | From [[Divisor Sum of Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$94 = 2 \times 47$
Hence:
{{... | Divisor Sum of Non-Square Semiprime/Examples/94/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/94/Proof_1 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition",
"Difference of Two Squares"
] |
proofwiki-12733 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | We have that:
:$94 = 2 \times 47$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {94}
| r = \paren {2 + 1} \paren {47 + 1}
| c = Divisor Sum of Non-Square Semiprime
}}
{{eqn | r = 3 \times 48
| c =
}}
{{eqn | r = 3 \times \paren ... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | We have that:
:$94 = 2 \times 47$
and so by definition is a [[Definition:Semiprime Number|semiprime]] whose [[Definition:Prime Factor|prime factors]] are [[Definition:Distinct|distinct]].
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {94}
| r = \paren {2 + 1} \paren {47 + 1}
| c = [[Divisor Sum of Non-... | Divisor Sum of Non-Square Semiprime/Examples/94/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Examples/94/Proof_2 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Semiprime Number",
"Definition:Prime Factor",
"Definition:Distinct",
"Divisor Sum of Non-Square Semiprime"
] |
proofwiki-12734 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | As $p$ and $q$ are distinct prime numbers, it follows that $p$ and $q$ are coprime.
Thus by Divisor Sum Function is Multiplicative:
:$\map {\sigma_1} n = \map {\sigma_1} p \map {\sigma_1} q$
From Divisor Sum of Prime Number:
:$\map {\sigma_1} p = \paren {p + 1}$
:$\map {\sigma_1} q = \paren {q + 1}$
Hence the result.
{... | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | As $p$ and $q$ are [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]], it follows that $p$ and $q$ are [[Definition:Coprime Integers|coprime]].
Thus by [[Divisor Sum Function is Multiplicative]]:
:$\map {\sigma_1} n = \map {\sigma_1} p \map {\sigma_1} q$
From [[Divisor Sum of Prime Number]]:
:... | Divisor Sum of Non-Square Semiprime/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Proof_1 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Distinct",
"Definition:Prime Number",
"Definition:Coprime/Integers",
"Divisor Sum Function is Multiplicative",
"Divisor Sum of Prime Number"
] |
proofwiki-12735 | Divisor Sum of Non-Square Semiprime | Let $n \in \Z_{>0}$ be a semiprime with distinct prime factors $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the divisor sum function. | A semiprime with distinct prime factors is a square-free integer.
By Divisor Sum of Square-Free Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} p_i + 1$
Hence the result.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] $p$ and $q$.
Then:
:$\map {\sigma_1} n = \paren {p + 1} \paren {q + 1}$
where $\map {\sigma_1} n$ denotes the [[Definition:Divisor Sum Function|divisor sum function]]. | A [[Definition:Semiprime Number|semiprime]] with [[Definition:Distinct|distinct]] [[Definition:Prime Factor|prime factors]] is a [[Definition:Square-Free|square-free]] [[Definition:Integer|integer]].
By [[Divisor Sum of Square-Free Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} p_i + 1$
He... | Divisor Sum of Non-Square Semiprime/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime | https://proofwiki.org/wiki/Divisor_Sum_of_Non-Square_Semiprime/Proof_2 | [
"Divisor Sum Function",
"Semiprimes",
"Divisor Sum of Non-Square Semiprime",
"Divisor Sum of Integer"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Divisor Sum Function"
] | [
"Definition:Semiprime Number",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Square-Free",
"Definition:Integer",
"Divisor Sum of Square-Free Integer"
] |
proofwiki-12736 | Integers which are Divisor Sum for 3 Integers | The sequence of integers which are the divisor sum of $3$ different integers begins:
:$24, 42, 48, 60, 84, 90, \ldots$
{{OEIS|A007372}} | For a given $n$, to determine every $m$ such that $\map {\sigma_1} m = n$ can be determined by evaluating the divisor sum of all integers up to $n - 1$.
It is hence noted:
{{begin-eqn}}
{{eqn | l = 24
| r = \map {\sigma_1} {14}
| c = {{DSFLink|14}}
}}
{{eqn | r = \map {\sigma_1} {15}
| c = {{DSFLink|1... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] which are the [[Definition:Divisor Sum Function|divisor sum]] of $3$ different [[Definition:Integer|integers]] begins:
:$24, 42, 48, 60, 84, 90, \ldots$
{{OEIS|A007372}} | For a given $n$, to determine every $m$ such that $\map {\sigma_1} m = n$ can be determined by evaluating the [[Definition:Divisor Sum Function|divisor sum]] of all [[Definition:Integer|integers]] up to $n - 1$.
It is hence noted:
{{begin-eqn}}
{{eqn | l = 24
| r = \map {\sigma_1} {14}
| c = {{DSFLink|14... | Integers which are Divisor Sum for 3 Integers | https://proofwiki.org/wiki/Integers_which_are_Divisor_Sum_for_3_Integers | https://proofwiki.org/wiki/Integers_which_are_Divisor_Sum_for_3_Integers | [
"Divisor Sum Function"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Divisor Sum Function",
"Definition:Integer"
] | [
"Definition:Divisor Sum Function",
"Definition:Integer",
"Divisor Sum of Prime Number",
"Definition:Prime Number",
"Divisor Sum of Prime Number",
"Definition:Prime Number",
"Divisor Sum of Prime Number",
"Definition:Prime Number",
"Divisor Sum of Prime Number",
"Definition:Prime Number",
"Diviso... |
proofwiki-12737 | Divisor Sum of Square-Free Integer | Let $n$ be an integer such that $n \ge 2$.
Let $n$ be square-free.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then:
:$\ds... | From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$66 = 2 \times 3 \times 11$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sig... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $n$ be [[Definition:Square-Free|square-free]].
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the [[Definition:Divisor... | From [[Divisor Sum of Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$66 = 2 \times 3 \times 11$
... | Divisor Sum of Square-Free Integer/Examples/66/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer/Examples/66/Proof_1 | [
"Divisor Sum Function",
"Divisor Sum of Integer",
"Divisor Sum of Square-Free Integer"
] | [
"Definition:Integer",
"Definition:Square-Free",
"Definition:Prime Decomposition",
"Definition:Divisor Sum Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition"
] |
proofwiki-12738 | Divisor Sum of Square-Free Integer | Let $n$ be an integer such that $n \ge 2$.
Let $n$ be square-free.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then:
:$\ds... | We have that:
:$66 = 2 \times 3 \times 11$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {66}
| r = \paren {2 + 1} \paren {3 + 1} \paren {11 + 1}
| c = Divisor Sum of Square-Free Integer
}}
{{eqn | r = 3 \times 4 \times 12
| c =
}}
{{eqn | r = 3 \times 2^2 \times \paren {2^2 \times 3}
| c = ... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $n$ be [[Definition:Square-Free|square-free]].
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the [[Definition:Divisor... | We have that:
:$66 = 2 \times 3 \times 11$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {66}
| r = \paren {2 + 1} \paren {3 + 1} \paren {11 + 1}
| c = [[Divisor Sum of Square-Free Integer]]
}}
{{eqn | r = 3 \times 4 \times 12
| c =
}}
{{eqn | r = 3 \times 2^2 \times \paren {2^2 \times 3}
... | Divisor Sum of Square-Free Integer/Examples/66/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer/Examples/66/Proof_2 | [
"Divisor Sum Function",
"Divisor Sum of Integer",
"Divisor Sum of Square-Free Integer"
] | [
"Definition:Integer",
"Definition:Square-Free",
"Definition:Prime Decomposition",
"Definition:Divisor Sum Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Sum of Square-Free Integer"
] |
proofwiki-12739 | Divisor Sum of Square-Free Integer | Let $n$ be an integer such that $n \ge 2$.
Let $n$ be square-free.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then:
:$\ds... | From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$70 = 2 \times 5 \times 7$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigm... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $n$ be [[Definition:Square-Free|square-free]].
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the [[Definition:Divisor... | From [[Divisor Sum of Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the [[Definition:Prime Decomposition|prime decomposition]] of $n$.
We have that:
:$70 = 2 \times 5 \times 7$
He... | Divisor Sum of Square-Free Integer/Examples/70/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer/Examples/70/Proof_1 | [
"Divisor Sum Function",
"Divisor Sum of Integer",
"Divisor Sum of Square-Free Integer"
] | [
"Definition:Integer",
"Definition:Square-Free",
"Definition:Prime Decomposition",
"Definition:Divisor Sum Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Sum of Integer",
"Definition:Prime Decomposition"
] |
proofwiki-12740 | Divisor Sum of Square-Free Integer | Let $n$ be an integer such that $n \ge 2$.
Let $n$ be square-free.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then:
:$\ds... | We have that:
:$70 = 2 \times 5 \times 7$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {70}
| r = \paren {2 + 1} \paren {5 + 1} \paren {7 + 1}
| c = Divisor Sum of Square-Free Integer
}}
{{eqn | r = 3 \times 6 \times 8
| c =
}}
{{eqn | r = 3 \times \paren {3 \times 2} \times 2^3
| c =
}}
{... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $n$ be [[Definition:Square-Free|square-free]].
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the [[Definition:Divisor... | We have that:
:$70 = 2 \times 5 \times 7$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {70}
| r = \paren {2 + 1} \paren {5 + 1} \paren {7 + 1}
| c = [[Divisor Sum of Square-Free Integer]]
}}
{{eqn | r = 3 \times 6 \times 8
| c =
}}
{{eqn | r = 3 \times \paren {3 \times 2} \times 2^3
| c =... | Divisor Sum of Square-Free Integer/Examples/70/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer/Examples/70/Proof_2 | [
"Divisor Sum Function",
"Divisor Sum of Integer",
"Divisor Sum of Square-Free Integer"
] | [
"Definition:Integer",
"Definition:Square-Free",
"Definition:Prime Decomposition",
"Definition:Divisor Sum Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Sum of Square-Free Integer"
] |
proofwiki-12741 | Divisor Sum of Square-Free Integer | Let $n$ be an integer such that $n \ge 2$.
Let $n$ be square-free.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then:
:$\ds... | We have that the Divisor Sum Function is Multiplicative.
From the definition of prime number, each of the prime factors of $n$ is coprime to all other divisors of $n$.
From Divisor Sum of Prime Number, we have:
:$\map {\sigma_1} {p_i} = p_i + 1$
Hence the result.
{{qed}} | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $n$ be [[Definition:Square-Free|square-free]].
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the [[Definition:Divisor... | We have that the [[Divisor Sum Function is Multiplicative]].
From the definition of [[Definition:Prime Number|prime number]], each of the [[Definition:Prime Factor|prime factors]] of $n$ is [[Definition:Coprime Integers|coprime]] to all other [[Definition:Divisor of Integer|divisors]] of $n$.
From [[Divisor Sum of Pr... | Divisor Sum of Square-Free Integer/Proof 1 | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer/Proof_1 | [
"Divisor Sum Function",
"Divisor Sum of Integer",
"Divisor Sum of Square-Free Integer"
] | [
"Definition:Integer",
"Definition:Square-Free",
"Definition:Prime Decomposition",
"Definition:Divisor Sum Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Sum Function is Multiplicative",
"Definition:Prime Number",
"Definition:Prime Factor",
"Definition:Coprime/Integers",
"Definition:Divisor (Algebra)/Integer",
"Divisor Sum of Prime Number"
] |
proofwiki-12742 | Divisor Sum of Square-Free Integer | Let $n$ be an integer such that $n \ge 2$.
Let $n$ be square-free.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then:
:$\ds... | From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where each of the $k_i$s are equal to $1$;
Hence:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^2 - 1} {p_i - 1}$
But from Difference of Two Squares:
:$\dfrac {p... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $n$ be [[Definition:Square-Free|square-free]].
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i = p_1 p_2 \cdots p_r$
Let $\map {\sigma_1} n$ be the [[Definition:Divisor... | From [[Divisor Sum of Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where each of the $k_i$s are equal to $1$;
Hence:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^2 - 1} {p_i - 1}$
But from [[Difference of Two Squares]]:
... | Divisor Sum of Square-Free Integer/Proof 2 | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer | https://proofwiki.org/wiki/Divisor_Sum_of_Square-Free_Integer/Proof_2 | [
"Divisor Sum Function",
"Divisor Sum of Integer",
"Divisor Sum of Square-Free Integer"
] | [
"Definition:Integer",
"Definition:Square-Free",
"Definition:Prime Decomposition",
"Definition:Divisor Sum Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Sum of Integer",
"Difference of Two Squares"
] |
proofwiki-12743 | Mapping Assigning to Element Its Lower Closure is Isomorphism | Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $I = \struct {\map {\mathit {Ids} } L, \precsim}$ be an inclusion ordered set
where:
:$\map {\mathit {Ids} } L$ denotes the set of all ideals in $L$
:$\mathord \precsim = \mathord \subseteq \cap \paren {\map {\mathit {Ids} } L \times \map {\m... | By definition:
:$\forall x \in S: x^\preceq$ is a principal ideal.
By Compact Element iff Principal Ideal:
:$\forall x \in S: x^\preceq$ is a compact element in $I$.
By definition of compact subset:
:$\forall x \in S: x^\preceq \in \map K I$
Then $f$ is well-defined.
We will prove that:
:$f$ is an order embedding.
That... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $I = \struct {\map {\mathit {Ids} } L, \precsim}$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]]
where:
:$\map {\mathit {Ids} } L$ denotes the [[... | By definition:
:$\forall x \in S: x^\preceq$ is a [[Definition:Principal Ideal of Preordered Set|principal ideal]].
By [[Compact Element iff Principal Ideal]]:
:$\forall x \in S: x^\preceq$ is a [[Definition:Compact Element|compact element]] in $I$.
By definition of [[Definition:Compact Subset of Lattice|compact subs... | Mapping Assigning to Element Its Lower Closure is Isomorphism | https://proofwiki.org/wiki/Mapping_Assigning_to_Element_Its_Lower_Closure_is_Isomorphism | https://proofwiki.org/wiki/Mapping_Assigning_to_Element_Its_Lower_Closure_is_Isomorphism | [
"Join and Meet Semilattices",
"Order Isomorphisms"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Definition:Ordered Subset",
"Definition:Compact Subset of Lattice",
"Definition:Mapping",
"Definition:Order Isomorphism"
] | [
"Definition:Principal Ideal of Preordered Set",
"Compact Element iff Principal Ideal",
"Definition:Compact Element",
"Definition:Compact Subset of Lattice",
"Definition:Order Embedding",
"Definition:Reflexivity",
"Definition:Lower Closure/Element",
"Definition:Subset",
"Definition:Lower Closure/Elem... |
proofwiki-12744 | Difference Triangle for Sequence of Fifth Powers | The difference triangle for the sequence of fifth powers ends on the fifth line with instances of $5!$, where $!$ denotes factorial. | {{ProofWanted|Needs some difference calculus results first.}} | The [[Definition:Difference Triangle|difference triangle]] for the [[Definition:Integer Sequence|sequence]] of [[Definition:Integer Power|fifth powers]] ends on the fifth line with instances of $5!$, where $!$ denotes [[Definition:Factorial|factorial]]. | {{ProofWanted|Needs some difference calculus results first.}} | Difference Triangle for Sequence of Fifth Powers | https://proofwiki.org/wiki/Difference_Triangle_for_Sequence_of_Fifth_Powers | https://proofwiki.org/wiki/Difference_Triangle_for_Sequence_of_Fifth_Powers | [
"Difference Calculus"
] | [
"Definition:Difference Triangle",
"Definition:Integer Sequence",
"Definition:Power (Algebra)/Integer",
"Definition:Factorial"
] | [] |
proofwiki-12745 | Ideals form Arithmetic Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below lattice.
Let $I = \struct {\map {\mathit {Ids} } L, \precsim}$ be an inclusion ordered set, where:
:$\map {\mathit {Ids} } L$ denotes the set of all ideals in $L$
:$\mathord \precsim = \mathord \subseteq \cap \paren {\map {\mathit {Ids} } L \times \map {\m... | We will prove that:
:$\forall x, y \in \map K I: \exists z \in \map K I: z \precsim x \land z \precsim y \land \forall v \in \map K I: v \precsim x \land v \precsim y \implies v \precsim z$
Let $x, y \in \map K I$.
By definition of compact subset:
:$x$ and $y$ are compact elements in $I$.
By Compact Element iff Princip... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Lattice (Order Theory)|lattice]].
Let $I = \struct {\map {\mathit {Ids} } L, \precsim}$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]], where:
:$\map {\mathit {Ids} } L$ denotes t... | We will prove that:
:$\forall x, y \in \map K I: \exists z \in \map K I: z \precsim x \land z \precsim y \land \forall v \in \map K I: v \precsim x \land v \precsim y \implies v \precsim z$
Let $x, y \in \map K I$.
By definition of [[Definition:Compact Subset of Lattice|compact subset]]:
:$x$ and $y$ are [[Definition... | Ideals form Arithmetic Lattice | https://proofwiki.org/wiki/Ideals_form_Arithmetic_Lattice | https://proofwiki.org/wiki/Ideals_form_Arithmetic_Lattice | [
"Lattice Theory",
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Lattice (Order Theory)",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Definition:Arithmetic Ordered Set",
"Definition:Lattice (Order Theory)"
] | [
"Definition:Compact Subset of Lattice",
"Definition:Compact Element",
"Compact Element iff Principal Ideal",
"Definition:Principal Ideal of Preordered Set",
"Intersection of Semilattice Ideals is Ideal",
"Meet in Set of Ideals",
"Definition:Lower Closure/Element",
"Meet Precedes Operands",
"Definiti... |
proofwiki-12746 | Sums of Partial Sequences of Squares | Let $n \in \Z_{>0}$.
Consider the odd number $2 n + 1$ and its square $\paren {2 n + 1}^2 = 2 m + 1$.
Then:
:$\ds \sum_{j \mathop = 0}^n \paren {m - j}^2 = \sum_{j \mathop = 1}^n \paren {m + j}^2$
That is:
:the sum of the squares of the $n + 1$ integers up to $m$
equals:
:the sum of the squares of the $n$ integers from... | First we express $m$ in terms of $n$:
{{begin-eqn}}
{{eqn | l = \paren {2 n + 1}^2
| r = 4 n^2 + 4 n + 1
| c =
}}
{{eqn | r = 2 \paren {2 n^2 + 2 n} + 1
| c =
}}
{{eqn | ll= \leadsto
| l = m
| r = 2 n^2 + 2 n
}}
{{end-eqn}}
We have:
{{begin-eqn}}
{{eqn | o =
| r = \sum_{j \mathop =... | Let $n \in \Z_{>0}$.
Consider the [[Definition:Odd Number|odd number]] $2 n + 1$ and its [[Definition:Square (Algebra)|square]] $\paren {2 n + 1}^2 = 2 m + 1$.
Then:
:$\ds \sum_{j \mathop = 0}^n \paren {m - j}^2 = \sum_{j \mathop = 1}^n \paren {m + j}^2$
That is:
:the [[Definition:Integer Addition|sum]] of the [[D... | First we express $m$ in terms of $n$:
{{begin-eqn}}
{{eqn | l = \paren {2 n + 1}^2
| r = 4 n^2 + 4 n + 1
| c =
}}
{{eqn | r = 2 \paren {2 n^2 + 2 n} + 1
| c =
}}
{{eqn | ll= \leadsto
| l = m
| r = 2 n^2 + 2 n
}}
{{end-eqn}}
We have:
{{begin-eqn}}
{{eqn | o =
| r = \sum_{j \matho... | Sums of Partial Sequences of Squares | https://proofwiki.org/wiki/Sums_of_Partial_Sequences_of_Squares | https://proofwiki.org/wiki/Sums_of_Partial_Sequences_of_Squares | [
"Sums of Sequences",
"Square Numbers",
"Sums of Partial Sequences of Squares"
] | [
"Definition:Odd Integer",
"Definition:Square/Function",
"Definition:Addition/Integers",
"Definition:Square/Function",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Square/Function",
"Definition:Integer"
] | [
"Difference of Two Squares",
"Closed Form for Triangular Numbers"
] |
proofwiki-12747 | Power of n equalling (n - 1)! + 1 | There is exactly one solution to the equation in the integers:
:$\paren {n - 1}! + 1 = n^k$
for $k > 1$, and that is:
:$n = 5$
:$k = 2$ | We have that:
{{begin-eqn}}
{{eqn | l = \paren {1 - 1}! + 1
| r = 0! + 1
| c =
}}
{{eqn | r = 1 + 1
| c = Factorial of Zero
}}
{{eqn | r = 2
| c = not a power of $1$
}}
{{eqn | l = \paren {2 - 1}! + 1
| r = 1! + 1
| c =
}}
{{eqn | r = 1 + 1
| c = Examples of Factorials
}}
{{e... | There is exactly one solution to the equation in the [[Definition:Integer|integers]]:
:$\paren {n - 1}! + 1 = n^k$
for $k > 1$, and that is:
:$n = 5$
:$k = 2$ | We have that:
{{begin-eqn}}
{{eqn | l = \paren {1 - 1}! + 1
| r = 0! + 1
| c =
}}
{{eqn | r = 1 + 1
| c = [[Factorial of Zero]]
}}
{{eqn | r = 2
| c = not a [[Definition:Integer Power|power]] of $1$
}}
{{eqn | l = \paren {2 - 1}! + 1
| r = 1! + 1
| c =
}}
{{eqn | r = 1 + 1
|... | Power of n equalling (n - 1)! + 1 | https://proofwiki.org/wiki/Power_of_n_equalling_(n_-_1)!_+_1 | https://proofwiki.org/wiki/Power_of_n_equalling_(n_-_1)!_+_1 | [
"Factorials"
] | [
"Definition:Integer"
] | [
"Factorial/Examples/0",
"Definition:Power (Algebra)/Integer",
"Factorial/Examples",
"Factorial/Examples",
"Factorial/Examples",
"Definition:Power (Algebra)/Integer",
"Factorial/Examples",
"Definition:Composite Number",
"Divisibility of n-1 Factorial by Composite n",
"Definition:Power (Algebra)/Int... |
proofwiki-12748 | Ideals form Algebraic Lattice | Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $I = \struct {\map {\operatorname{Ids} } L, \precsim}$ be an inclusion ordered set
where
:$\map {\operatorname{Ids} } L$ denotes the set of all ideals in $L$
:$\mathord \precsim = \mathord \subseteq \cap \paren {\map {\operatorname{Ids} } L \... | By definition of subset:
:$\map {\operatorname{Ids} } L \subseteq \powerset S$
where $\powerset S$ denotes the power set of $S$.
Define:
:$P = \struct {\powerset S, \precsim'}$
where:
:$\mathord \precsim' = \mathord\subseteq \cap \paren {\powerset S \times \powerset S}$
By Ideals are Continuous Lattice Subframe of Powe... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $I = \struct {\map {\operatorname{Ids} } L, \precsim}$ be an [[Definition:Subset|inclusion]] [[Definition:Ordered Set|ordered set]]
where
:$\map {\operatorname{Ids} } L$ denot... | By definition of [[Definition:Subset|subset]]:
:$\map {\operatorname{Ids} } L \subseteq \powerset S$
where $\powerset S$ denotes the [[Definition:Power Set|power set]] of $S$.
Define:
:$P = \struct {\powerset S, \precsim'}$
where:
:$\mathord \precsim' = \mathord\subseteq \cap \paren {\powerset S \times \powerset S}$
... | Ideals form Algebraic Lattice | https://proofwiki.org/wiki/Ideals_form_Algebraic_Lattice | https://proofwiki.org/wiki/Ideals_form_Algebraic_Lattice | [
"Join and Meet Semilattices",
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)"
] | [
"Definition:Subset",
"Definition:Power Set",
"Ideals are Continuous Lattice Subframe of Power Set",
"Definition:Continuous Lattice Subframe",
"Lattice of Power Set is Algebraic",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Continuous Lattice Subframe of Algebraic Lattice ... |
proofwiki-12749 | Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite | Let
:$\map P x : \closedint a b \to \R$
:$\map h x : \closedint a b \to \R$.
Let $\map h x$ be continuously differentiable $\forall x \in \closedint a b$.
Suppose:
:$\forall x \in \closedint a b: \map P x > 0$
Then:
:$\ds \forall \map h x : \map h a = \map h b = 0 : \int_a^b \paren {P h'^2 + Q h^2} \rd x > 0$
{{iff}} ... | === Necessary Condition ===
Let $\map \omega x : \closedint a b \to \R$ be a continuously differentiable mapping.
Then:
{{begin-eqn}}
{{eqn | l = 0
| r = \bigintlimits {\omega h^2} a b
| c = Boundary Conditions for $h$
}}
{{eqn | r = \int_a^b \map {\frac \d {\d x} } {\omega h^2} \rd x
| c = Fundamenta... | Let
:$\map P x : \closedint a b \to \R$
:$\map h x : \closedint a b \to \R$.
Let $\map h x$ be [[Definition:Continuously Differentiable|continuously differentiable]] $\forall x \in \closedint a b$.
Suppose:
:$\forall x \in \closedint a b: \map P x > 0$
Then:
:$\ds \forall \map h x : \map h a = \map h b = 0 : ... | === Necessary Condition ===
Let $\map \omega x : \closedint a b \to \R$ be a [[Definition:Continuously Differentiable Real Function|continuously differentiable mapping]].
Then:
{{begin-eqn}}
{{eqn | l = 0
| r = \bigintlimits {\omega h^2} a b
| c = [[Definition:Boundary Condition|Boundary Conditions]] for... | Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Quadratic_Functional_to_be_Positive_Definite | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Quadratic_Functional_to_be_Positive_Definite | [
"Calculus of Variations"
] | [
"Definition:Continuously Differentiable",
"Definition:Interval/Ordered Set/Closed",
"Definition:Point",
"Definition:Conjugate Point (Calculus of Variations)"
] | [
"Definition:Continuously Differentiable/Real Function",
"Definition:Boundary Condition",
"Fundamental Theorem of Calculus/Second Part",
"Definition:Differential Equation/Solution",
"Definition:Differential Equation",
"Existence-Uniqueness Theorem for First-Order Differential Equation",
"Definition:Infin... |
proofwiki-12750 | Square Cullen Numbers | The numbers:
:$1, 9, 25$
are Cullen numbers which are also square. | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = 0 \times 2^0 + 1
}}
{{eqn | l = 9
| r = 2 \times 2^2 + 1
}}
{{eqn | l = 25
| r = 3 \times 2^3 + 1
}}
{{end-eqn}}
{{Qed}} | The numbers:
:$1, 9, 25$
are [[Definition:Cullen Number|Cullen numbers]] which are also [[Definition:Square Number|square]]. | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = 0 \times 2^0 + 1
}}
{{eqn | l = 9
| r = 2 \times 2^2 + 1
}}
{{eqn | l = 25
| r = 3 \times 2^3 + 1
}}
{{end-eqn}}
{{Qed}} | Square Cullen Numbers | https://proofwiki.org/wiki/Square_Cullen_Numbers | https://proofwiki.org/wiki/Square_Cullen_Numbers | [
"Cullen Numbers",
"Square Numbers"
] | [
"Definition:Cullen Number",
"Definition:Square Number"
] | [] |
proofwiki-12751 | Smallest Non-Palindromic Number with Palindromic Square | $26$ is the smallest non-palindromic integer whose square is palindromic. | Checking the squares of all non-palindromic integers in turn from $10$ upwards, until a palindromic integer is reached:
{{begin-eqn}}
{{eqn | l = 10^2
| r = 100
}}
{{eqn | l = 12^2
| r = 144
}}
{{eqn | l = 13^2
| r = 169
}}
{{eqn | l = 14^2
| r = 196
}}
{{eqn | l = 15^2
| r = 225
}}
{{eqn ... | $26$ is the smallest non-[[Definition:Palindromic Number|palindromic integer]] whose [[Definition:Square Number|square]] is [[Definition:Palindromic Number|palindromic]]. | Checking the [[Definition:Square Number|squares]] of all non-[[Definition:Palindromic Number|palindromic integers]] in turn from $10$ upwards, until a [[Definition:Palindromic Number|palindromic integer]] is reached:
{{begin-eqn}}
{{eqn | l = 10^2
| r = 100
}}
{{eqn | l = 12^2
| r = 144
}}
{{eqn | l = 13^2... | Smallest Non-Palindromic Number with Palindromic Square | https://proofwiki.org/wiki/Smallest_Non-Palindromic_Number_with_Palindromic_Square | https://proofwiki.org/wiki/Smallest_Non-Palindromic_Number_with_Palindromic_Square | [
"26",
"Square Numbers",
"Palindromic Numbers"
] | [
"Definition:Palindromic Number",
"Definition:Square Number",
"Definition:Palindromic Number"
] | [
"Definition:Square Number",
"Definition:Palindromic Number",
"Definition:Palindromic Number"
] |
proofwiki-12752 | Numbers Partitioned into Six Hexagonal Numbers | The integers $11$ and $26$ cannot be represented by the sum of less than $6$ hexagonal numbers. | Recall the sequence of hexagonal numbers:
{{:Hexagonal Number/Sequence}}
Hence:
{{begin-eqn}}
{{eqn | l = 11
| r = 6 + 1 + 1 + 1 + 1 + 1
| c =
}}
{{eqn | l = 26
| r = 6 + 6 + 6 + 6 + 1 + 1
| c =
}}
{{end-eqn}}
{{qed}} | The [[Definition:Integer|integers]] $11$ and $26$ cannot be represented by the [[Definition:Integer Addition|sum]] of less than $6$ [[Definition:Hexagonal Number|hexagonal numbers]]. | Recall the [[Hexagonal Number/Sequence|sequence of hexagonal numbers]]:
{{:Hexagonal Number/Sequence}}
Hence:
{{begin-eqn}}
{{eqn | l = 11
| r = 6 + 1 + 1 + 1 + 1 + 1
| c =
}}
{{eqn | l = 26
| r = 6 + 6 + 6 + 6 + 1 + 1
| c =
}}
{{end-eqn}}
{{qed}} | Numbers Partitioned into Six Hexagonal Numbers | https://proofwiki.org/wiki/Numbers_Partitioned_into_Six_Hexagonal_Numbers | https://proofwiki.org/wiki/Numbers_Partitioned_into_Six_Hexagonal_Numbers | [
"11",
"26",
"Hexagonal Numbers"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Hexagonal Number"
] | [
"Hexagonal Number/Sequence"
] |
proofwiki-12753 | Integer as Sum of 27 Primes | Every positive integer greater than $1$ can be expressed as the sum of no more than $27$ primes. | {{questionable|A far cry from the Goldbach Conjecture. Puzzling statement.}} | Every [[Definition:Positive Integer|positive integer]] greater than $1$ can be expressed as the [[Definition:Integer Addition|sum]] of no more than $27$ [[Definition:Prime Number|primes]]. | {{questionable|A far cry from the [[Goldbach Conjecture]]. Puzzling statement.}} | Integer as Sum of 27 Primes | https://proofwiki.org/wiki/Integer_as_Sum_of_27_Primes | https://proofwiki.org/wiki/Integer_as_Sum_of_27_Primes | [
"Prime Numbers",
"27"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Prime Number"
] | [
"Goldbach Conjecture"
] |
proofwiki-12754 | Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice.
Let $P = \struct {T, \precsim}$ be a continuous lattice subframe of $L$.
Then $P$ is algebraic lattice. | By definition of algebraic ordered set:
:$L$ is up-complete.
By Up-Complete Lower Bounded Join Semilattice is Complete:
:$L$ is a complete lattice.
By definition:
:$P$ is closure system of $L$.
By Image of Operator Generated by Closure System is Set of Closure System
:$\map {\operatorname {operator} } P \sqbrk S = T$
B... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice (Order Theory)|lattice]].
Let $P = \struct {T, \precsim}$ be a [[Definition:Continuous Lattice Subframe|continuous lattice subframe]] of $L$.
Then $P$ i... | By definition of [[Definition:Algebraic Ordered Set|algebraic ordered set]]:
:$L$ is [[Definition:Up-Complete|up-complete]].
By [[Up-Complete Lower Bounded Join Semilattice is Complete]]:
:$L$ is a [[Definition:Complete Lattice|complete lattice]].
By definition:
:$P$ is [[Definition:Closure System|closure system]] of... | Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice | https://proofwiki.org/wiki/Continuous_Lattice_Subframe_of_Algebraic_Lattice_is_Algebraic_Lattice | https://proofwiki.org/wiki/Continuous_Lattice_Subframe_of_Algebraic_Lattice_is_Algebraic_Lattice | [
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Continuous Lattice Subframe",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)"
] | [
"Definition:Algebraic Ordered Set",
"Definition:Up-Complete",
"Up-Complete Lower Bounded Join Semilattice is Complete",
"Definition:Complete Lattice",
"Definition:Closure System",
"Image of Operator Generated by Closure System is Set of Closure System",
"Closure Operator Preserves Directed Suprema iff I... |
proofwiki-12755 | 27 is Smallest Number whose Period of Reciprocal is 3 | $27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$:
:$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$ | From Reciprocal of $27$:
{{:Reciprocal of 27}}
It can be determined by inspection of all smaller integers that this is indeed the smallest to have a period of $3$.
{{qed}} | $27$ is the smallest [[Definition:Positive Integer|positive integer]] the [[Definition:Decimal Expansion|decimal expansion]] of whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $3$:
:$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$ | From [[Reciprocal of 27|Reciprocal of $27$]]:
{{:Reciprocal of 27}}
It can be determined by inspection of all smaller integers that this is indeed the smallest to have a [[Definition:Period of Recurrence|period]] of $3$.
{{qed}} | 27 is Smallest Number whose Period of Reciprocal is 3 | https://proofwiki.org/wiki/27_is_Smallest_Number_whose_Period_of_Reciprocal_is_3 | https://proofwiki.org/wiki/27_is_Smallest_Number_whose_Period_of_Reciprocal_is_3 | [
"27",
"Examples of Reciprocals"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 27",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-12756 | Long Period Prime/Examples/7 | $7$ is the smallest long period prime:
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$ | From Reciprocal of $7$:
{{:Reciprocal of 7}}
The reciprocals of $1$, $2$, $4$ and $5$ do not recur:
{{begin-eqn}}
{{eqn | l = \frac 1 1
| r = 1
| c =
}}
{{eqn | l = \frac 1 2
| r = 0 \cdotp 5
| c =
}}
{{eqn | l = \frac 1 4
| r = 0 \cdotp 25
| c =
}}
{{eqn | l = \frac 1 5
| r... | $7$ is the smallest [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$ | From [[Reciprocal of 7|Reciprocal of $7$]]:
{{:Reciprocal of 7}}
The [[Definition:Reciprocal|reciprocals]] of $1$, $2$, $4$ and $5$ do not [[Definition:Recurrence of Basis Expansion|recur]]:
{{begin-eqn}}
{{eqn | l = \frac 1 1
| r = 1
| c =
}}
{{eqn | l = \frac 1 2
| r = 0 \cdotp 5
| c =
}}
... | Long Period Prime/Examples/7 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/7 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/7 | [
"7",
"Examples of Long Period Primes"
] | [
"Definition:Long Period Prime"
] | [
"Reciprocal of 7",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence",
"Definition:Basis Expansion/Recurrence",
"Definition:Basis Expansion/Recurrence/Period"
] |
proofwiki-12757 | Sequence of Smallest Numbers whose Reciprocal has Period n | Let $\sequence {s_n}$ be the sequence defined as:
:$s_n$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $n$
for $n = 0, 1, 2, \ldots$
Then $\sequence {s_n}$ begins:
:$1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, \ldots$
{{OEIS|A003060}} | Demonstrated by inspection and calculation:
{{begin-eqn}}
{{eqn | l = \frac 1 1
| r = 1 \cdotp 0
| c = hence has a period of $0$
}}
{{eqn | l = \frac 1 3
| r = 0 \cdotp \dot 3
| c = Reciprocal of $3$
}}
{{eqn | l = \frac 1 {11}
| r = 0 \cdotp \dot 0 \dot 9
| c = Reciprocal of $11$
}}... | Let $\sequence {s_n}$ be the [[Definition:Integer Sequence|sequence]] defined as:
:$s_n$ is the smallest [[Definition:Positive Integer|positive integer]] the [[Definition:Decimal Expansion|decimal expansion]] of whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $n$
for $n = ... | Demonstrated by inspection and calculation:
{{begin-eqn}}
{{eqn | l = \frac 1 1
| r = 1 \cdotp 0
| c = hence has a [[Definition:Period of Recurrence|period]] of $0$
}}
{{eqn | l = \frac 1 3
| r = 0 \cdotp \dot 3
| c = [[Reciprocal of 3|Reciprocal of $3$]]
}}
{{eqn | l = \frac 1 {11}
| r =... | Sequence of Smallest Numbers whose Reciprocal has Period n | https://proofwiki.org/wiki/Sequence_of_Smallest_Numbers_whose_Reciprocal_has_Period_n | https://proofwiki.org/wiki/Sequence_of_Smallest_Numbers_whose_Reciprocal_has_Period_n | [
"Reciprocals"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Definition:Basis Expansion/Recurrence/Period",
"Reciprocal of 3",
"Reciprocal of 11",
"Reciprocal of 27",
"Reciprocal of 101",
"Reciprocal of 41",
"Reciprocal of 7",
"Reciprocal of 239",
"Reciprocal of 73",
"Reciprocal of 81",
"Reciprocal of 451"
] |
proofwiki-12758 | Numbers whose Cyclic Permutations of 3-Digit Multiples are Multiples | Let $n$ be a two-digit positive integer with the following property:
:Let $m$ be a $3$-digit multiple of $n$.
:Then any cyclic permutation of the digits of $m$ is also a multiple of $n$.
Then $n$ is either $27$ or $37$. | Let $m$ be a multiple of $n$ with $3$ digits.
Then we have:
{{begin-eqn}}
{{eqn | l = n \times c
| r = a_2 \times 10^2 + a_1 \times 10^1 + a_0
}}
{{end-eqn}}
Let us now cyclically permute the digits of $m$ by multiplying by $10$.
Then we have:
{{begin-eqn}}
{{eqn | l = 10 \times n \times c
| r = 10 \times ... | Let $n$ be a two-[[Definition:Digit|digit]] [[Definition:Positive Integer|positive integer]] with the following property:
:Let $m$ be a $3$-[[Definition:Digit|digit]] [[Definition:Multiple of Integer|multiple]] of $n$.
:Then any [[Definition:Cyclic Permutation|cyclic permutation]] of the [[Definition:Digit|digits]] o... | Let $m$ be a [[Definition:Multiple of Integer|multiple]] of $n$ with $3$ [[Definition:Digit|digits]].
Then we have:
{{begin-eqn}}
{{eqn | l = n \times c
| r = a_2 \times 10^2 + a_1 \times 10^1 + a_0
}}
{{end-eqn}}
Let us now [[Definition:Cyclic Permutation|cyclically permute]] the [[Definition:Digit|digits]] ... | Numbers whose Cyclic Permutations of 3-Digit Multiples are Multiples | https://proofwiki.org/wiki/Numbers_whose_Cyclic_Permutations_of_3-Digit_Multiples_are_Multiples | https://proofwiki.org/wiki/Numbers_whose_Cyclic_Permutations_of_3-Digit_Multiples_are_Multiples | [
"Number Theory",
"27",
"37"
] | [
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Multiple/Integer",
"Definition:Cyclic Permutation",
"Definition:Digit",
"Definition:Multiple/Integer"
] | [
"Definition:Multiple/Integer",
"Definition:Digit",
"Definition:Cyclic Permutation",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Cyclic Permutation",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12759 | Sequence of Successive Longest Collatz Sequence Generators | The sequence of integers which generate a Collatz process which is longer than that of any smaller integers begins:
{{begin-eqn}}
{{eqn | l = 1
| o = :
| c = $0$ steps
}}
{{eqn | l = 2
| o = :
| c = $1$ step
}}
{{eqn | l = 3
| o = :
| c = $7$ steps
}}
{{eqn | l = 6
| o = :
... | Missing complementary orbits:
{{begin-eqn}}
{{eqn | l = 4
| o = :
| c = $2$ steps
}}
{{eqn | l = 5
| o = :
| c = $5$ steps
}}
{{eqn | l = 8
| o = :
| c = $3$ steps
}}
{{eqn | l = 10
| o = :
| c = $6$ steps
}}
{{end-eqn}}
{{Proofread}} | The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] which generate a [[Collatz Conjecture|Collatz process]] which is longer than that of any smaller [[Definition:Integer|integers]] begins:
{{begin-eqn}}
{{eqn | l = 1
| o = :
| c = $0$ steps
}}
{{eqn | l = 2
| o = :
|... | Missing complementary orbits:
{{begin-eqn}}
{{eqn | l = 4
| o = :
| c = $2$ steps
}}
{{eqn | l = 5
| o = :
| c = $5$ steps
}}
{{eqn | l = 8
| o = :
| c = $3$ steps
}}
{{eqn | l = 10
| o = :
| c = $6$ steps
}}
{{end-eqn}}
{{Proofread}} | Sequence of Successive Longest Collatz Sequence Generators | https://proofwiki.org/wiki/Sequence_of_Successive_Longest_Collatz_Sequence_Generators | https://proofwiki.org/wiki/Sequence_of_Successive_Longest_Collatz_Sequence_Generators | [
"Collatz Conjecture"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Collatz Conjecture",
"Definition:Integer"
] | [] |
proofwiki-12760 | Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice.
Let $c: S \to S$ be a closure operator that preserves directed suprema.
Let $C = \struct {c \sqbrk S, \precsim}$ be an ordered subset of $L$.
Then $C$ is algebraic lattice. | By definition of algebraic ordered set:
:$L$ is up-complete.
By Up-Complete Lower Bounded Join Semilattice is Complete:
:$L$ is a complete lattice.
By definition of closure operator:
:$c$ is idempotent.
By Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice:
:$C$ is a complete lattice.
We wi... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Algebraic Ordered Set|algebraic]] [[Definition:Lattice (Order Theory)|lattice]].
Let $c: S \to S$ be a [[Definition:Closure Operator|closure operator]] that [[Definition:Mapping Preserves Supremum/Directed|pre... | By definition of [[Definition:Algebraic Ordered Set|algebraic ordered set]]:
:$L$ is [[Definition:Up-Complete|up-complete]].
By [[Up-Complete Lower Bounded Join Semilattice is Complete]]:
:$L$ is a [[Definition:Complete Lattice|complete lattice]].
By definition of [[Definition:Closure Operator|closure operator]]:
:$c... | Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice | https://proofwiki.org/wiki/Image_of_Directed_Suprema_Preserving_Closure_Operator_is_Algebraic_Lattice | https://proofwiki.org/wiki/Image_of_Directed_Suprema_Preserving_Closure_Operator_is_Algebraic_Lattice | [
"Continuous Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Closure Operator",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Ordered Subset",
"Definition:Algebraic Ordered Set",
"Definition:Lattice (Order Theory)"
] | [
"Definition:Algebraic Ordered Set",
"Definition:Up-Complete",
"Up-Complete Lower Bounded Join Semilattice is Complete",
"Definition:Complete Lattice",
"Definition:Closure Operator",
"Definition:Idempotence/Mapping",
"Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice",
"De... |
proofwiki-12761 | Boundary of Compact Closed Set is Compact | Let $X$ be a topological space.
Let $K\subset X$ be a compact subspace of $X$.
Let $K$ be closed in $X$.
Then its boundary $\partial K$ is compact. | By Boundary of Set is Closed, $\partial K$ is closed in $X$.
By Set is Closed iff it Contains its Boundary, $\partial K \subset K$.
By Closed Set in Topological Subspace, $\partial K$ is closed in $K$.
By Closed Subspace of Compact Space is Compact, $\partial K$ is compact.
{{qed}} | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $K\subset X$ be a [[Definition:Compact Topological Subspace|compact subspace]] of $X$.
Let $K$ be [[Definition:Closed Set (Topology)|closed]] in $X$.
Then its [[Definition:Boundary (Topology)|boundary]] $\partial K$ is [[Definition:Compact Topolog... | By [[Boundary of Set is Closed]], $\partial K$ is [[Definition:Closed Set (Topology)|closed]] in $X$.
By [[Set is Closed iff it Contains its Boundary]], $\partial K \subset K$.
By [[Closed Set in Topological Subspace]], $\partial K$ is [[Definition:Closed Set (Topology)|closed]] in $K$.
By [[Closed Subspace of Compa... | Boundary of Compact Closed Set is Compact | https://proofwiki.org/wiki/Boundary_of_Compact_Closed_Set_is_Compact | https://proofwiki.org/wiki/Boundary_of_Compact_Closed_Set_is_Compact | [
"Set Boundaries",
"Compact Topological Spaces",
"Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Closed Set/Topology",
"Definition:Boundary (Topology)",
"Definition:Compact Topological Space"
] | [
"Boundary of Set is Closed",
"Definition:Closed Set/Topology",
"Set is Closed iff it Contains its Boundary",
"Closed Set in Topological Subspace",
"Definition:Closed Set/Topology",
"Closed Subspace of Compact Space is Compact",
"Definition:Compact Topological Space"
] |
proofwiki-12762 | Discrete Subgroup of Hausdorff Group is Closed | Let $G$ be a Hausdorff topological group.
Let $H$ be a discrete subgroup of $G$.
Then $H$ is closed in $G$. | Let $g \in \overline H$ be in the closure of $H$.
We will show that $g \in H$.
{{AimForCont}} $g \notin H$.
Let $e$ be the identity of $G$.
Because $H$ is discrete, there exists an open set $U \subset G$ such that $U \cap H = \set e$.
Then $V = U \cap U^{-1}$ is an open neighborhood of $e$ in $G$.
By Right and Left Reg... | Let $G$ be a [[Definition:Hausdorff Space|Hausdorff]] [[Definition:Topological Group|topological group]].
Let $H$ be a [[Definition:Discrete Subgroup|discrete subgroup]] of $G$.
Then $H$ is [[Definition:Closed Set (Topology)|closed]] in $G$. | Let $g \in \overline H$ be in the [[Definition:Closure (Topology)|closure]] of $H$.
We will show that $g \in H$.
{{AimForCont}} $g \notin H$.
Let $e$ be the [[Definition:Identity Element|identity]] of $G$.
Because $H$ is [[Definition:Discrete Space|discrete]], there exists an [[Definition:Open Set (Topology)|open s... | Discrete Subgroup of Hausdorff Group is Closed | https://proofwiki.org/wiki/Discrete_Subgroup_of_Hausdorff_Group_is_Closed | https://proofwiki.org/wiki/Discrete_Subgroup_of_Hausdorff_Group_is_Closed | [
"Topological Groups",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Topological Group",
"Definition:Discrete Subgroup",
"Definition:Closed Set/Topology"
] | [
"Definition:Closure (Topology)",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Discrete Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Neighborhood (Topology)/Point",
"Right and Left Regular Representations in Topological Group are Homeo... |
proofwiki-12763 | Boundary of Compact Set in Hausdorff Space is Compact | Let $T = \struct {S, \tau}$ be a Hausdorff topological space.
Let $K \subset S$ be a compact subspace of $T$.
Then its boundary $\partial K$ is compact. | By Compact Subspace of Hausdorff Space is Closed, $K$ is closed in $T$.
By Boundary of Compact Closed Set is Compact, $\partial K$ is compact.
{{qed}}
Category:Set Boundaries
Category:Compact Topological Spaces
Category:Closed Sets
Category:Hausdorff Spaces
07a2hroov7qiiahtgttwg0xpq8tzk2t | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff]] [[Definition:Topological Space|topological space]].
Let $K \subset S$ be a [[Definition:Compact Topological Subspace|compact subspace]] of $T$.
Then its [[Definition:Boundary (Topology)|boundary]] $\partial K$ is [[Definition:Compact Topologic... | By [[Compact Subspace of Hausdorff Space is Closed]], $K$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
By [[Boundary of Compact Closed Set is Compact]], $\partial K$ is [[Definition:Compact Topological Space|compact]].
{{qed}}
[[Category:Set Boundaries]]
[[Category:Compact Topological Spaces]]
[[Category:Cl... | Boundary of Compact Set in Hausdorff Space is Compact | https://proofwiki.org/wiki/Boundary_of_Compact_Set_in_Hausdorff_Space_is_Compact | https://proofwiki.org/wiki/Boundary_of_Compact_Set_in_Hausdorff_Space_is_Compact | [
"Set Boundaries",
"Compact Topological Spaces",
"Closed Sets",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Topological Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Boundary (Topology)",
"Definition:Compact Topological Space"
] | [
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Closed Set/Topology",
"Boundary of Compact Closed Set is Compact",
"Definition:Compact Topological Space",
"Category:Set Boundaries",
"Category:Compact Topological Spaces",
"Category:Closed Sets",
"Category:Hausdorff Spaces"
] |
proofwiki-12764 | Group is Hausdorff iff Discrete Subgroups are Closed | A topological group is Hausdorff {{iff}} its discrete subgroups are closed. | Follows directly from:
:Discrete Subgroup of Hausdorff Group is Closed
:Topological Group is Hausdorff iff Identity is Closed
{{qed}} | A [[Definition:Topological Group|topological group]] is [[Definition:Hausdorff Space|Hausdorff]] {{iff}} its [[Definition:Discrete Subgroup|discrete subgroups]] are [[Definition:Closed Set (Topology)|closed]]. | Follows directly from:
:[[Discrete Subgroup of Hausdorff Group is Closed]]
:[[Topological Group is Hausdorff iff Identity is Closed]]
{{qed}} | Group is Hausdorff iff Discrete Subgroups are Closed | https://proofwiki.org/wiki/Group_is_Hausdorff_iff_Discrete_Subgroups_are_Closed | https://proofwiki.org/wiki/Group_is_Hausdorff_iff_Discrete_Subgroups_are_Closed | [
"Topological Groups",
"Hausdorff Spaces"
] | [
"Definition:Topological Group",
"Definition:T2 Space",
"Definition:Discrete Subgroup",
"Definition:Closed Set/Topology"
] | [
"Discrete Subgroup of Hausdorff Group is Closed",
"Topological Group is Hausdorff iff Identity is Closed"
] |
proofwiki-12765 | Group is Hausdorff iff has Closed Discrete Subgroup | A topological group is Hausdorff {{iff}} it has a closed discrete subgroup. | === Necessary Condition ===
Follows directly from Topological Group is Hausdorff iff Identity is Closed.
{{explain|How?}}
{{qed|lemma}} | A [[Definition:Topological Group|topological group]] is [[Definition:Hausdorff Space|Hausdorff]] {{iff}} it has a [[Definition:Closed Set (Topology)|closed]] [[Definition:Discrete Subgroup|discrete subgroup]]. | === Necessary Condition ===
Follows directly from [[Topological Group is Hausdorff iff Identity is Closed]].
{{explain|How?}}
{{qed|lemma}} | Group is Hausdorff iff has Closed Discrete Subgroup | https://proofwiki.org/wiki/Group_is_Hausdorff_iff_has_Closed_Discrete_Subgroup | https://proofwiki.org/wiki/Group_is_Hausdorff_iff_has_Closed_Discrete_Subgroup | [
"Topological Groups",
"Hausdorff Spaces"
] | [
"Definition:Topological Group",
"Definition:T2 Space",
"Definition:Closed Set/Topology",
"Definition:Discrete Subgroup"
] | [
"Topological Group is Hausdorff iff Identity is Closed",
"Topological Group is Hausdorff iff Identity is Closed"
] |
proofwiki-12766 | Closure of Subgroup is Group | Let $G$ be a topological group.
Let $H\leq G$ be a subgroup.
Let $\overline H$ denote its closure.
Then $\overline H$ is a subgroup of $G$. | We use the One-Step Subgroup Test.
Because $H \subset \overline H$, $\overline H$ is non-empty.
Let $a, b \in \overline H$.
Let $U$ be a neighborhood of $a b^{-1}$.
Let the mapping $f: G\times G \to G$ be defined as:
:$\map f {x, y} = x y^{-1}$
By definition of topological group, $f$ is continuous.
By definition of pro... | Let $G$ be a [[Definition:Topological Group|topological group]].
Let $H\leq G$ be a [[Definition:Subgroup|subgroup]].
Let $\overline H$ denote its [[Definition:Closure (Topology)|closure]].
Then $\overline H$ is a [[Definition:Subgroup|subgroup]] of $G$. | We use the [[One-Step Subgroup Test]].
Because $H \subset \overline H$, $\overline H$ is [[Definition:Non-Empty Set|non-empty]].
Let $a, b \in \overline H$.
Let $U$ be a [[Definition:Neighborhood of Point|neighborhood]] of $a b^{-1}$.
Let the [[Definition:Mapping|mapping]] $f: G\times G \to G$ be defined as:
:$\map... | Closure of Subgroup is Group | https://proofwiki.org/wiki/Closure_of_Subgroup_is_Group | https://proofwiki.org/wiki/Closure_of_Subgroup_is_Group | [
"Topological Groups"
] | [
"Definition:Topological Group",
"Definition:Subgroup",
"Definition:Closure (Topology)",
"Definition:Subgroup"
] | [
"One-Step Subgroup Test",
"Definition:Non-Empty Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Mapping",
"Definition:Topological Group",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Neighborhood (Topology)/Point",
"On... |
proofwiki-12767 | Image of Group Homomorphism is Hausdorff Implies Kernel is Closed | Let $G$ and $H$ be topological groups.
Let $f: G \to H$ be a morphism.
Let its image $\Img f$ be Hausdorff.
Then its kernel $\map \ker f$ is closed in $G$. | By Image of Group Homomorphism is Subgroup, $\Img f$ is a group.
Let $e$ be the identity of $H$.
By Topological Group is Hausdorff iff Identity is Closed, $\set e$ is closed in $\Img f$.
Because $f$ is continuous, $\map \ker f = \map {f^{-1} } e$ is closed in $G$.
{{qed}}
Category:Topological Groups
Category:Hausdorff ... | Let $G$ and $H$ be [[Definition:Topological Group|topological groups]].
Let $f: G \to H$ be a [[Definition:Topological Group Homomorphism|morphism]].
Let its [[Definition:Image of Mapping|image]] $\Img f$ be [[Definition:Hausdorff Space|Hausdorff]].
Then its [[Definition:Kernel of Group Homomorphism|kernel]] $\map ... | By [[Image of Group Homomorphism is Subgroup]], $\Img f$ is a [[Definition:Group|group]].
Let $e$ be the [[Definition:Identity Element|identity]] of $H$.
By [[Topological Group is Hausdorff iff Identity is Closed]], $\set e$ is [[Definition:Closed Set (Topology)|closed]] in $\Img f$.
Because $f$ is [[Definition:Cont... | Image of Group Homomorphism is Hausdorff Implies Kernel is Closed | https://proofwiki.org/wiki/Image_of_Group_Homomorphism_is_Hausdorff_Implies_Kernel_is_Closed | https://proofwiki.org/wiki/Image_of_Group_Homomorphism_is_Hausdorff_Implies_Kernel_is_Closed | [
"Topological Groups",
"Hausdorff Spaces"
] | [
"Definition:Topological Group",
"Definition:Topological Group Homomorphism",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:T2 Space",
"Definition:Kernel of Group Homomorphism",
"Definition:Closed Set/Topology"
] | [
"Image of Group Homomorphism is Subgroup",
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Topological Group is Hausdorff iff Identity is Closed",
"Definition:Closed Set/Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Closed Set/Topology",
"Category... |
proofwiki-12768 | Group is Connected iff Subgroup and Quotient are Connected | Let $G$ be a topological group.
Let $H \le G$ be a subgroup.
{{TFAE}}
:$(1): \quad G$ is connected
:$(2): \quad H$ is connected and the left quotient space $G / H$ is connected
:$(3): \quad H$ is connected and the right quotient space $G / H$ is connected. | {{ProofWanted}}
Category:Topological Groups
Category:Connected Topological Spaces
Category:Quotient Spaces (Topology)
99kmfihh1whhdxtg8jiqafyg9130q7z | Let $G$ be a [[Definition:Topological Group|topological group]].
Let $H \le G$ be a [[Definition:Subgroup|subgroup]].
{{TFAE}}
:$(1): \quad G$ is [[Definition:Connected Topological Space|connected]]
:$(2): \quad H$ is [[Definition:Connected Topological Space|connected]] and the [[Definition:Left Coset Space|left quo... | {{ProofWanted}}
[[Category:Topological Groups]]
[[Category:Connected Topological Spaces]]
[[Category:Quotient Spaces (Topology)]]
99kmfihh1whhdxtg8jiqafyg9130q7z | Group is Connected iff Subgroup and Quotient are Connected | https://proofwiki.org/wiki/Group_is_Connected_iff_Subgroup_and_Quotient_are_Connected | https://proofwiki.org/wiki/Group_is_Connected_iff_Subgroup_and_Quotient_are_Connected | [
"Topological Groups",
"Connected Topological Spaces",
"Quotient Spaces (Topology)"
] | [
"Definition:Topological Group",
"Definition:Subgroup",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space",
"Definition:Coset Space/Left Coset Space",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space",
"Definition:Coset Space/Right Coset ... | [
"Category:Topological Groups",
"Category:Connected Topological Spaces",
"Category:Quotient Spaces (Topology)"
] |
proofwiki-12769 | Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open | Let $G$ be a group acting by homeomorphisms on a topological space $X$.
Then the projection map $\pi: X \to X / G$ is open. | Let $U \subset X$ be open.
We have to show that $\pi \sqbrk U$ is open.
By definition of quotient topology, this is the case {{iff}} $\pi^{-1} \sqbrk {\pi \sqbrk U}$ is open.
By definition of saturation under group action:
:$\ds \pi^{-1} \sqbrk {\pi \sqbrk U} = \bigcup_{g \mathop \in G} g U$
Because $G$ acts by homeomo... | Let $G$ be a [[Definition:Group Action by Homeomorphisms|group acting by homeomorphisms]] on a [[Definition:Topological Space|topological space]] $X$.
Then the [[Definition:Quotient Mapping|projection map]] $\pi: X \to X / G$ is [[Definition:Open Mapping|open]]. | Let $U \subset X$ be [[Definition:Open Set (Topology)|open]].
We have to show that $\pi \sqbrk U$ is [[Definition:Open Set (Topology)|open]].
By definition of [[Definition:Quotient Topology|quotient topology]], this is the case {{iff}} $\pi^{-1} \sqbrk {\pi \sqbrk U}$ is [[Definition:Open Set (Topology)|open]].
By d... | Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open | https://proofwiki.org/wiki/Group_Acts_by_Homeomorphisms_Implies_Projection_on_Quotient_Space_is_Open | https://proofwiki.org/wiki/Group_Acts_by_Homeomorphisms_Implies_Projection_on_Quotient_Space_is_Open | [
"Topology",
"Group Actions"
] | [
"Definition:Group Action by Homeomorphisms",
"Definition:Topological Space",
"Definition:Quotient Mapping",
"Definition:Open Mapping"
] | [
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Quotient Topology",
"Definition:Open Set/Topology",
"Definition:Saturation (Group Action)",
"Definition:Group Action by Homeomorphisms",
"Definition:Open Set/Topology"
] |
proofwiki-12770 | Neighborhood iff Contains Neighborhood | Let $X$ be a topological space.
Let $x\in X$.
Let $V\subset X$ be a subset.
Then the following are equivalent:
:$V$ is a neighborhood of $x$ in $X$
:$V$ contains a neighborhood of $x$ in $X$ | Follows directly from the definition of neighborhood and Subset Relation is Transitive.
{{qed}} | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $x\in X$.
Let $V\subset X$ be a [[Definition:Subset|subset]].
Then the following are [[Definition:Logical Equivalence|equivalent]]:
:$V$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$ in $X$
:$V$ contains a [[Definition:Neighborhood... | Follows directly from the definition of [[Definition:Neighborhood of Point|neighborhood]] and [[Subset Relation is Transitive]].
{{qed}} | Neighborhood iff Contains Neighborhood | https://proofwiki.org/wiki/Neighborhood_iff_Contains_Neighborhood | https://proofwiki.org/wiki/Neighborhood_iff_Contains_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Logical Equivalence",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point",
"Subset Relation is Transitive"
] |
proofwiki-12771 | Even Perfect Number ends in 6 or 28 preceded by Odd Digit | Let $n$ be an even perfect number.
Then $n$ ends either in $6$ or $28$ preceded by an odd digit. | By the Theorem of Even Perfect Numbers:
:$n = 2^{p - 1} \paren {2^p - 1}$
where $p$ is prime.
With the exception of $6 = 2^1 \paren {2^2 - 1}$ and $28 = 2^2 \paren {2^3 - 1}$:
:$p$ is odd and $p > 4$.
We claim that:
:$n$ ends in $\phantom 0 6$ preceded by an odd digit if $p \equiv 1 \pmod 4$
:$n$ ends in $28$ preceded ... | Let $n$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]].
Then $n$ ends either in $6$ or $28$ preceded by an [[Definition:Odd Integer|odd]] [[Definition:Digit|digit]]. | By the [[Theorem of Even Perfect Numbers]]:
:$n = 2^{p - 1} \paren {2^p - 1}$
where $p$ is [[Definition:Prime Number|prime]].
With the exception of $6 = 2^1 \paren {2^2 - 1}$ and $28 = 2^2 \paren {2^3 - 1}$:
:$p$ is [[Definition:Odd Integer|odd]] and $p > 4$.
We claim that:
:$n$ ends in $\phantom 0 6$ preceded by an... | Even Perfect Number ends in 6 or 28 preceded by Odd Digit | https://proofwiki.org/wiki/Even_Perfect_Number_ends_in_6_or_28_preceded_by_Odd_Digit | https://proofwiki.org/wiki/Even_Perfect_Number_ends_in_6_or_28_preceded_by_Odd_Digit | [
"Perfect Numbers"
] | [
"Definition:Even Integer",
"Definition:Perfect Number",
"Definition:Odd Integer",
"Definition:Digit"
] | [
"Theorem of Even Perfect Numbers",
"Definition:Prime Number",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Digit",
"Definition:Odd Integer",
"Definition:Digit",
"Powers of 16 Modulo 20",
"Powers of 16 Modulo 20",
"Powers of 16 Modulo 20"
] |
proofwiki-12772 | Open Subset of Locally Connected Space is Locally Connected | Let $T = \struct {S, \tau}$ be a locally connected topological space.
Let $U \subseteq S$ be open in $T$.
Then $U$ is locally connected. | {{MissingLinks}} | Let $T = \struct {S, \tau}$ be a [[Definition:Locally Connected Space|locally connected]] [[Definition:Topological Space|topological space]].
Let $U \subseteq S$ be [[Definition:Open Set (Topology)|open]] in $T$.
Then $U$ is [[Definition:Locally Connected Space|locally connected]]. | {{MissingLinks}} | Open Subset of Locally Connected Space is Locally Connected | https://proofwiki.org/wiki/Open_Subset_of_Locally_Connected_Space_is_Locally_Connected | https://proofwiki.org/wiki/Open_Subset_of_Locally_Connected_Space_is_Locally_Connected | [
"Open Subset of Locally Connected Space is Locally Connected",
"Topology",
"Locally Connected Spaces"
] | [
"Definition:Locally Connected Space",
"Definition:Topological Space",
"Definition:Open Set/Topology",
"Definition:Locally Connected Space"
] | [] |
proofwiki-12773 | Component of Locally Connected Space is Open | Let $T = \left({S, \tau}\right)$ be a locally connected topological space.
Let $G$ be a component of $T$.
Then $G$ is open. | By definition of locally connected space, $T$ has a basis of connected sets in $T$.
Thus $S$ is a union of open connected sets in $T$.
By Components are Open iff Union of Open Connected Sets, the components of $T$ are open.
{{qed}} | Let $T = \left({S, \tau}\right)$ be a [[Definition:Locally Connected Space|locally connected]] [[Definition:Topological Space|topological space]].
Let $G$ be a [[Definition:Component (Topology)|component]] of $T$.
Then $G$ is [[Definition:Open Set (Topology)|open]]. | By definition of [[Definition:Locally Connected Space/Definition 3|locally connected space]], $T$ has a [[Definition:Basis (Topology)|basis]] of [[Definition:Connected Set (Topology)|connected sets]] in $T$.
Thus $S$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Connected ... | Component of Locally Connected Space is Open | https://proofwiki.org/wiki/Component_of_Locally_Connected_Space_is_Open | https://proofwiki.org/wiki/Component_of_Locally_Connected_Space_is_Open | [
"Components (Topology)",
"Locally Connected Spaces"
] | [
"Definition:Locally Connected Space",
"Definition:Topological Space",
"Definition:Component (Topology)",
"Definition:Open Set/Topology"
] | [
"Definition:Locally Connected Space/Definition 3",
"Definition:Basis (Topology)",
"Definition:Connected Set (Topology)",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Connected Set (Topology)",
"Components are Open iff Union of Open Connected Sets",
"Definition:Component (Topolog... |
proofwiki-12774 | Path Component of Locally Path-Connected Space is Open | Let $T = \struct {S, \tau}$ be a locally path-connected topological space.
Let $G$ be a path component of $T$.
Then $G$ is open in $T$. | By definition of locally path-connected, $T$ has a basis of path-connected set.
Thus $S$ is a union of open path-connected sets of $T$.
By Path Components are Open iff Union of Open Path-Connected Sets, the path components of $T$ are open in $T$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Locally Path-Connected Space|locally path-connected]] [[Definition:Topological Space|topological space]].
Let $G$ be a [[Definition:Path Component|path component]] of $T$.
Then $G$ is [[Definition:Open Set (Topology)|open]] in $T$. | By definition of [[Definition:Locally Path-Connected Space|locally path-connected]], $T$ has a [[Definition:Basis (Topology)|basis]] of [[Definition:Path-Connected Set|path-connected set]].
Thus $S$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Path-Connected Set|path-conn... | Path Component of Locally Path-Connected Space is Open | https://proofwiki.org/wiki/Path_Component_of_Locally_Path-Connected_Space_is_Open | https://proofwiki.org/wiki/Path_Component_of_Locally_Path-Connected_Space_is_Open | [
"Path Components",
"Locally Path-Connected Spaces"
] | [
"Definition:Locally Path-Connected Space",
"Definition:Topological Space",
"Definition:Path Component",
"Definition:Open Set/Topology"
] | [
"Definition:Locally Path-Connected Space",
"Definition:Basis (Topology)",
"Definition:Path-Connected/Set",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Path-Connected/Set",
"Path Components are Open iff Union of Open Path-Connected Sets",
"Definition:Path Component",
"Definiti... |
proofwiki-12775 | Connected and Locally Path-Connected Implies Path-Connected | Let $T = \struct {S, \tau}$ be a connected and locally path-connected topological space.
Then $T$ is path-connected. | By:
:Path Component of Locally Path-Connected Space is Closed
:Path Component of Locally Path-Connected Space is Open
the path components of $T$ are clopen.
Because $T$ is connected, every path component equals $S$.
That is, $T$ is path-connected.
{{qed}}
Category:Connected Topological Spaces
Category:Path-Connected Sp... | Let $T = \struct {S, \tau}$ be a [[Definition:Connected Topological Space|connected]] and [[Definition:Locally Path-Connected Space|locally path-connected]] [[Definition:Topological Space|topological space]].
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | By:
:[[Path Component of Locally Path-Connected Space is Closed]]
:[[Path Component of Locally Path-Connected Space is Open]]
the [[Definition:Path Component|path components]] of $T$ are [[Definition:Clopen Set|clopen]].
Because $T$ is [[Definition:Connected Topological Space|connected]], every [[Definition:Path Comp... | Connected and Locally Path-Connected Implies Path-Connected | https://proofwiki.org/wiki/Connected_and_Locally_Path-Connected_Implies_Path-Connected | https://proofwiki.org/wiki/Connected_and_Locally_Path-Connected_Implies_Path-Connected | [
"Connected Topological Spaces",
"Path-Connected Spaces",
"Locally Path-Connected Spaces"
] | [
"Definition:Connected Topological Space",
"Definition:Locally Path-Connected Space",
"Definition:Topological Space",
"Definition:Path-Connected/Topological Space"
] | [
"Path Component of Locally Path-Connected Space is Closed",
"Path Component of Locally Path-Connected Space is Open",
"Definition:Path Component",
"Definition:Clopen Set",
"Definition:Connected Topological Space",
"Definition:Path Component",
"Definition:Path-Connected/Topological Space",
"Category:Co... |
proofwiki-12776 | Components are Open iff Union of Open Connected Sets | Let $T = \struct {S, \tau}$ be a topological space.
{{TFAE}}
:$(1): \quad$ The components of $T$ are open.
:$(2): \quad S$ is a union of open connected sets of $T$. | === Condition (1) implies Condition (2) ===
{{:Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets}}{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
{{TFAE}}
:$(1): \quad$ The [[Definition:Component (Topology)|components]] of $T$ are [[Definition:Open Set (Topology)|open]].
:$(2): \quad S$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definiti... | === [[Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets|Condition (1) implies Condition (2)]] ===
{{:Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets}}{{qed|lemma}} | Components are Open iff Union of Open Connected Sets | https://proofwiki.org/wiki/Components_are_Open_iff_Union_of_Open_Connected_Sets | https://proofwiki.org/wiki/Components_are_Open_iff_Union_of_Open_Connected_Sets | [
"Components are Open iff Union of Open Connected Sets",
"Connected Topological Spaces",
"Components (Topology)"
] | [
"Definition:Topological Space",
"Definition:Component (Topology)",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Connected Set (Topology)"
] | [
"Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets"
] |
proofwiki-12777 | Path Components are Open iff Union of Open Path-Connected Sets | Let $T = \struct {S, \tau}$ be a topological space.
{{TFAE}}
:$(1): \quad$ The path components of $T$ are open.
:$(2): \quad S$ is a union of open path-connected sets of $T$. | === Condition (1) implies Condition (2) ===
{{:Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets}}{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
{{TFAE}}
:$(1): \quad$ The [[Definition:Path Component|path components]] of $T$ are [[Definition:Open Set (Topology)|open]].
:$(2): \quad S$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definitio... | === [[Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets|Condition (1) implies Condition (2)]] ===
{{:Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connec... | Path Components are Open iff Union of Open Path-Connected Sets | https://proofwiki.org/wiki/Path_Components_are_Open_iff_Union_of_Open_Path-Connected_Sets | https://proofwiki.org/wiki/Path_Components_are_Open_iff_Union_of_Open_Path-Connected_Sets | [
"Path Components are Open iff Union of Open Path-Connected Sets",
"Path-Connected Spaces",
"Path Components"
] | [
"Definition:Topological Space",
"Definition:Path Component",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Path-Connected/Set"
] | [
"Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets"
] |
proofwiki-12778 | Open Subset of Locally Path-Connected Space is Locally Path-Connected | Let $T = \struct {S, \tau}$ be a locally path-connected topological space.
Let $U \subset S$ be open in $T$.
Then $U$ is locally path-connected in $T$. | {{tidy}}
{{MissingLinks}}
Let $\tau_U$ denotes the subspace topology on $U$ induced by $\tau$.
That is, $\tau_U = \set {\OO \cap U: \OO \in \tau}$.
To show that $\struct {U, \tau_U}$ is a locally path-connected topological space, we must prove that each point $x \in U$ has a local basis consisting of path-connected set... | Let $T = \struct {S, \tau}$ be a [[Definition:Locally Path-Connected Space|locally path-connected]] [[Definition:Topological Space|topological space]].
Let $U \subset S$ be [[Definition:Open Set (Topology)|open]] in $T$.
Then $U$ is [[Definition:Locally Path-Connected Space|locally path-connected]] in $T$. | {{tidy}}
{{MissingLinks}}
Let $\tau_U$ denotes the [[Definition:Topological Subspace|subspace topology]] on $U$ induced by $\tau$.
That is, $\tau_U = \set {\OO \cap U: \OO \in \tau}$.
To show that $\struct {U, \tau_U}$ is a [[Definition:Locally Path-Connected Space|locally path-connected]] topological space, we must... | Open Subset of Locally Path-Connected Space is Locally Path-Connected | https://proofwiki.org/wiki/Open_Subset_of_Locally_Path-Connected_Space_is_Locally_Path-Connected | https://proofwiki.org/wiki/Open_Subset_of_Locally_Path-Connected_Space_is_Locally_Path-Connected | [
"Locally Path-Connected Spaces"
] | [
"Definition:Locally Path-Connected Space",
"Definition:Topological Space",
"Definition:Open Set/Topology",
"Definition:Locally Path-Connected Space"
] | [
"Definition:Topological Subspace",
"Definition:Locally Path-Connected Space",
"Definition:Element",
"Definition:Local Basis",
"Definition:Path-Connected/Set",
"Definition:Locally Path-Connected Space",
"Definition:Local Basis",
"Definition:Path-Connected/Set",
"Definition:Local Basis",
"Definition... |
proofwiki-12779 | Even Perfect Number is Hexagonal | All perfect numbers which are even are hexagonal. | Let $a$ be an even perfect number.
From the Theorem of Even Perfect Numbers, $a$ is in the form $2^{p - 1} \paren {2^p - 1}$ where $2^p - 1$ is prime.
Thus:
{{begin-eqn}}
{{eqn | l = a
| r = \paren {2^p - 1} 2^{p - 1}
| c =
}}
{{eqn | r = 2^{p - 1} \paren {2 \times 2^{p - 1} - 1}
| c =
}}
{{eqn | r... | All [[Definition:Perfect Number|perfect numbers]] which are [[Definition:Even Integer|even]] are [[Definition:Hexagonal Number|hexagonal]]. | Let $a$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]].
From the [[Theorem of Even Perfect Numbers]], $a$ is in the form $2^{p - 1} \paren {2^p - 1}$ where $2^p - 1$ is [[Definition:Prime Number|prime]].
Thus:
{{begin-eqn}}
{{eqn | l = a
| r = \paren {2^p - 1} 2^{p - 1}
... | Even Perfect Number is Hexagonal | https://proofwiki.org/wiki/Even_Perfect_Number_is_Hexagonal | https://proofwiki.org/wiki/Even_Perfect_Number_is_Hexagonal | [
"Euclidean Numbers",
"Hexagonal Numbers",
"Perfect Numbers"
] | [
"Definition:Perfect Number",
"Definition:Even Integer",
"Definition:Hexagonal Number"
] | [
"Definition:Even Integer",
"Definition:Perfect Number",
"Theorem of Even Perfect Numbers",
"Definition:Prime Number",
"Closed Form for Hexagonal Numbers"
] |
proofwiki-12780 | Hexagonal Number is Triangular Number | Let $H_n$ be the $n$th hexagonal number.
Then $H_n$ is the $2 n - 1$th triangular number. | {{begin-eqn}}
{{eqn | l = H_n
| r = n \paren {2 n - 1}
| c = Closed Form for Hexagonal Numbers
}}
{{eqn | r = \frac {2 n \paren {2 n - 1} } 2
| c =
}}
{{eqn | r = \frac {m \paren {m + 1} } 2
| c = for $m = 2 n - 1$
}}
{{end-eqn}}
Hence the result.
{{qed}} | Let $H_n$ be the $n$th [[Definition:Hexagonal Number|hexagonal number]].
Then $H_n$ is the $2 n - 1$th [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = H_n
| r = n \paren {2 n - 1}
| c = [[Closed Form for Hexagonal Numbers]]
}}
{{eqn | r = \frac {2 n \paren {2 n - 1} } 2
| c =
}}
{{eqn | r = \frac {m \paren {m + 1} } 2
| c = for $m = 2 n - 1$
}}
{{end-eqn}}
Hence the result.
{{qed}} | Hexagonal Number is Triangular Number | https://proofwiki.org/wiki/Hexagonal_Number_is_Triangular_Number | https://proofwiki.org/wiki/Hexagonal_Number_is_Triangular_Number | [
"Hexagonal Numbers",
"Triangular Numbers"
] | [
"Definition:Hexagonal Number",
"Definition:Triangular Number"
] | [
"Closed Form for Hexagonal Numbers"
] |
proofwiki-12781 | Perfect Number which is Sum of Equal Powers of Two Numbers | $28$ is the only perfect number which is the sum of equal powers of exactly $2$ positive integers:
:$28 = 1^3 + 3^3$ | Let $N$ be an even perfect number which is the sum of equal powers of exactly $2$ positive integers.
By Theorem of Even Perfect Numbers, write:
:$N = 2^{p - 1} \paren {2^p - 1} = a^n + b^n$
where $p, 2^p - 1$ are prime, $a, b, n \in \Z_{>0}$, $n > 1$.
{{WLOG}}, let $a \le b$.
Immediately we see that $a \ne b$:
{{AimFor... | $28$ is the only [[Definition:Perfect Number|perfect number]] which is the [[Definition:Integer Addition|sum]] of equal [[Definition:Integer Power|powers]] of exactly $2$ [[Definition:Positive Integer|positive integers]]:
:$28 = 1^3 + 3^3$ | Let $N$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]] which is the [[Definition:Integer Addition|sum]] of equal [[Definition:Integer Power|powers]] of exactly $2$ [[Definition:Positive Integer|positive integers]].
By [[Theorem of Even Perfect Numbers]], write:
:$N = 2^{p - 1} \par... | Perfect Number which is Sum of Equal Powers of Two Numbers | https://proofwiki.org/wiki/Perfect_Number_which_is_Sum_of_Equal_Powers_of_Two_Numbers | https://proofwiki.org/wiki/Perfect_Number_which_is_Sum_of_Equal_Powers_of_Two_Numbers | [
"Perfect Numbers",
"28"
] | [
"Definition:Perfect Number",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Positive/Integer"
] | [
"Definition:Even Integer",
"Definition:Perfect Number",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Positive/Integer",
"Theorem of Even Perfect Numbers",
"Definition:Prime Number",
"Prime Divides Power",
"Definition:Contradiction",
"Integer as Sum of Two Squar... |
proofwiki-12782 | Sum of Reciprocals of Divisors of Perfect Number is 2 | Let $n$ be a perfect number.
Then:
:$\ds \sum_{d \mathop \divides n} \dfrac 1 d = 2$
That is, the sum of the reciprocals of the divisors of $n$ is equal to $2$. | {{begin-eqn}}
{{eqn | l = \sum_{d \mathop \divides n} d
| r = \map {\sigma_1} n
| c = {{Defof|Divisor Sum Function}}
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 n \sum_{d \mathop \divides n} d
| r = \dfrac {\map {\sigma_1} n} n
| c =
}}
{{eqn | ll= \leadsto
| l = \sum_{d \mathop \divides... | Let $n$ be a [[Definition:Perfect Number|perfect number]].
Then:
:$\ds \sum_{d \mathop \divides n} \dfrac 1 d = 2$
That is, the [[Definition:Integer Addition|sum]] of the [[Definition:Reciprocal|reciprocals]] of the [[Definition:Divisor of Integer|divisors]] of $n$ is equal to $2$. | {{begin-eqn}}
{{eqn | l = \sum_{d \mathop \divides n} d
| r = \map {\sigma_1} n
| c = {{Defof|Divisor Sum Function}}
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 n \sum_{d \mathop \divides n} d
| r = \dfrac {\map {\sigma_1} n} n
| c =
}}
{{eqn | ll= \leadsto
| l = \sum_{d \mathop \divides... | Sum of Reciprocals of Divisors of Perfect Number is 2 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Divisors_of_Perfect_Number_is_2 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Divisors_of_Perfect_Number_is_2 | [
"Perfect Numbers"
] | [
"Definition:Perfect Number",
"Definition:Addition/Integers",
"Definition:Reciprocal",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Perfect Number/Definition 4"
] |
proofwiki-12783 | Even Perfect Number except 6 is Congruent to 1 Modulo 9 | Let $n$ be an even perfect number, but not $6$.
Then:
:$n \equiv 1 \pmod 9$ | From Theorem of Even Perfect Numbers:
:$n = 2^{p - 1} \paren {2^p - 1} = \dfrac {2^p \paren {2^p - 1} } 2$
where $p$ is prime.
From Odd Power of 2 is Congruent to 2 Modulo 3:
:$2^p \equiv 2 \pmod 3$
for odd $p$.
Thus:
{{begin-eqn}}
{{eqn | l = n
| r = \dfrac {\paren {3 k + 2} \paren {3 k + 1} } 2
| c = for ... | Let $n$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]], but not $6$.
Then:
:$n \equiv 1 \pmod 9$ | From [[Theorem of Even Perfect Numbers]]:
:$n = 2^{p - 1} \paren {2^p - 1} = \dfrac {2^p \paren {2^p - 1} } 2$
where $p$ is [[Definition:Prime Number|prime]].
From [[Odd Power of 2 is Congruent to 2 Modulo 3]]:
:$2^p \equiv 2 \pmod 3$
for [[Definition:Odd Prime|odd $p$]].
Thus:
{{begin-eqn}}
{{eqn | l = n
|... | Even Perfect Number except 6 is Congruent to 1 Modulo 9 | https://proofwiki.org/wiki/Even_Perfect_Number_except_6_is_Congruent_to_1_Modulo_9 | https://proofwiki.org/wiki/Even_Perfect_Number_except_6_is_Congruent_to_1_Modulo_9 | [
"Euclidean Numbers",
"Perfect Numbers"
] | [
"Definition:Even Integer",
"Definition:Perfect Number"
] | [
"Theorem of Even Perfect Numbers",
"Definition:Prime Number",
"Odd Power of 2 is Congruent to 2 Modulo 3",
"Definition:Odd Prime",
"Closed Form for Triangular Numbers",
"Definition:Triangular Number"
] |
proofwiki-12784 | Congruence Modulo 3 of Power of 2 | Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$2^n \equiv \paren {-1}^n \pmod 3$
where $\equiv$ denotes congruence.
That is:
:$\exists q \in \Z: 2^n = 3 q + \paren {-1}^n$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$2^n \equiv \paren {-1}^n \pmod 3$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = 2^0
| r = 1
| c =
}}
{{eqn | r = \paren {-1}^0
| c =
}}
{{eqn | o = \equiv
| r = \paren {-1}^0
| rr= \pmod... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$2^n \equiv \paren {-1}^n \pmod 3$
where $\equiv$ denotes [[Definition:Congruence Modulo Integer|congruence]].
That is:
:$\exists q \in \Z: 2^n = 3 q + \paren {-1}^n$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$2^n \equiv \paren {-1}^n \pmod 3$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = 2^0
| r = 1
| c =
}}
{{eqn | r = \paren {-1}^0
|... | Congruence Modulo 3 of Power of 2 | https://proofwiki.org/wiki/Congruence_Modulo_3_of_Power_of_2 | https://proofwiki.org/wiki/Congruence_Modulo_3_of_Power_of_2 | [
"Examples of Congruence Modulo Integer",
"2",
"3"
] | [
"Definition:Positive/Integer",
"Definition:Congruence (Number Theory)/Integers"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12785 | Odd Power of 2 is Congruent to 2 Modulo 3 | Let $n \in \Z_{\ge 0}$ be an odd positive integer.
Then:
:$2^n \equiv 2 \pmod 3$ | From Congruence Modulo 3 of Power of 2:
:$2^n \equiv \paren {-1}^n \pmod 3$
We have that $n$ is odd.
Hence:
{{begin-eqn}}
{{eqn | l = 2^n
| o = \equiv
| r = -1
| rr= \pmod 3
| c =
}}
{{eqn | o = \equiv
| r = 3 - 1
| rr= \pmod 3
| c =
}}
{{eqn | o = \equiv
| r = 2
... | Let $n \in \Z_{\ge 0}$ be an [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integer]].
Then:
:$2^n \equiv 2 \pmod 3$ | From [[Congruence Modulo 3 of Power of 2]]:
:$2^n \equiv \paren {-1}^n \pmod 3$
We have that $n$ is [[Definition:Odd Integer|odd]].
Hence:
{{begin-eqn}}
{{eqn | l = 2^n
| o = \equiv
| r = -1
| rr= \pmod 3
| c =
}}
{{eqn | o = \equiv
| r = 3 - 1
| rr= \pmod 3
| c =
}}
{{eqn... | Odd Power of 2 is Congruent to 2 Modulo 3 | https://proofwiki.org/wiki/Odd_Power_of_2_is_Congruent_to_2_Modulo_3 | https://proofwiki.org/wiki/Odd_Power_of_2_is_Congruent_to_2_Modulo_3 | [
"Number Theory",
"2",
"3"
] | [
"Definition:Odd Integer",
"Definition:Positive/Integer"
] | [
"Congruence Modulo 3 of Power of 2",
"Definition:Odd Integer",
"Category:Number Theory",
"Category:2",
"Category:3"
] |
proofwiki-12786 | Open Set in Open Subspace | Let $T = \struct{X, \tau}$ be a topological space.
Let $T_U = \struct{U, \tau_U}$ be a subspace of $T$ where $U \subseteq X$ is open.
Let $V \subseteq U$ be a subset.
Then $V$ is open in $T_U$ {{iff}} $V$ is open in $T$. | === Necessary Condition ===
Let $V$ be open in $T$.
By Intersection with Subset is Subset, $V\cap U = V$.
By definition of topological subspace, $V$ is open in $T_U$. | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T_U = \struct{U, \tau_U}$ be a [[Definition:Topological Subspace|subspace]] of $T$ where $U \subseteq X$ is [[Definition:Open Set (Topology)|open]].
Let $V \subseteq U$ be a [[Definition:Subset|subset]].
Then $V$ is [[Definitio... | === Necessary Condition ===
Let $V$ be [[Definition:Open Set (Topology)|open]] in $T$.
By [[Intersection with Subset is Subset]], $V\cap U = V$.
By definition of [[Definition:Topological Subspace|topological subspace]], $V$ is [[Definition:Open Set (Topology)|open]] in $T_U$. | Open Set in Open Subspace | https://proofwiki.org/wiki/Open_Set_in_Open_Subspace | https://proofwiki.org/wiki/Open_Set_in_Open_Subspace | [
"Topology",
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Topological Subspace",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] | [
"Definition:Open Set/Topology",
"Intersection with Subset is Subset",
"Definition:Topological Subspace",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Topological Subspace",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-12787 | Equivalence of Definitions of Locally Compact Hausdorff Space | {{TFAE|def = Locally Compact Hausdorff Space|view = locally compact Hausdorff space}}
Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space. | === $1$ implies $2$ ===
Let $x \in S$.
Let $K$ be a compact neighborhood of $x$.
Let $\BB$ be the set of compact neighborhoods of $x$.
It is shown that $\BB$ is a neighborhood basis of $x$.
Let $U$ be a neighborhood of $x$.
We have to show that $U$ contains a compact neighborhood of $x$.
By Neighborhood in Topological ... | {{TFAE|def = Locally Compact Hausdorff Space|view = locally compact Hausdorff space}}
Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | === $1$ implies $2$ ===
Let $x \in S$.
Let $K$ be a [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood of Point|neighborhood]] of $x$.
Let $\BB$ be the [[Definition:Set|set]] of [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood of Point|neighborhoods]] of $x$.
... | Equivalence of Definitions of Locally Compact Hausdorff Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Compact_Hausdorff_Space | [
"Locally Compact Hausdorff Spaces"
] | [
"Definition:T2 Space"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:Set",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood Basis",
"Definition:Neighborhood (Topology)/Point",
"Definition:Compact T... |
proofwiki-12788 | Neighborhood in Compact Hausdorff Space Contains Compact Neighborhood | Let $X$ be a compact Hausdorff topological space.
Let $x\in X$.
Let $U$ be a neighborhood of $x$.
Then $U$ contains a compact neighborhood of $x$. | By definition of neighborhood, there exists an open set $V$ with $x\in V\subset U$.
Then $X \setminus V$ is closed.
By Compact Hausdorff Space is $T_4$, there exist disjoint open sets $A, B$ such that:
:$X \setminus V \subset A$
:$x \in B$
Then:
:$X \setminus A$ is compact by Closed Subspace of Compact Space is Compact... | Let $X$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff]] [[Definition:Topological Space|topological space]].
Let $x\in X$.
Let $U$ be a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
Then $U$ contains a [[Definition:Compact Subspace|compact]] [[Definition:Nei... | By definition of [[Definition:Neighborhood of Point|neighborhood]], there exists an [[Definition:Open Set (Topology)|open set]] $V$ with $x\in V\subset U$.
Then $X \setminus V$ is [[Definition:Closed Set (Topology)|closed]].
By [[Compact Hausdorff Space is T4|Compact Hausdorff Space is $T_4$]], there exist [[Definiti... | Neighborhood in Compact Hausdorff Space Contains Compact Neighborhood | https://proofwiki.org/wiki/Neighborhood_in_Compact_Hausdorff_Space_Contains_Compact_Neighborhood | https://proofwiki.org/wiki/Neighborhood_in_Compact_Hausdorff_Space_Contains_Compact_Neighborhood | [
"Compact Topological Spaces",
"Hausdorff Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Topological Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Compact Hausdorff Space is T4",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Compact Topological Space/Subspace",
"Closed Subspace of Compact Space is Compact",
"... |
proofwiki-12789 | Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $f:S \to S$ be a mapping that is idempotent and preserves directed suprema.
Let $F = \struct {f \sqbrk S, \precsim}$ be an ordered subset of $L$.
Then $F$ inherits directed suprema and is complete lattice. | We will prove that
:$F$ inherits directed suprema.
Let $D$ be a directed subset of $f \sqbrk S$ such that
:$D$ admits a supremum in $L$.
By definition of ordered subset:
:$D$ is directed in $L$.
By definition of mapping preserves directed suprema:
:$\map {\sup_L} {f \sqbrk D} = \map f {\sup_L D}$
By definition of idemp... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $f:S \to S$ be a [[Definition:Mapping|mapping]] that is [[Definition:Idempotent Mapping|idempotent]] and [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
Let $F = \struct {f \sqbrk S, ... | We will prove that
:$F$ [[Definition:Directed Suprema Inheriting|inherits directed suprema]].
Let $D$ be a [[Definition:Directed Subset|directed subset]] of $f \sqbrk S$ such that
:$D$ admits a [[Definition:Supremum of Set|supremum]] in $L$.
By definition of [[Definition:Ordered Subset|ordered subset]]:
:$D$ is [[Def... | Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice | https://proofwiki.org/wiki/Image_of_Idempotent_and_Directed_Suprema_Preserving_Mapping_is_Complete_Lattice | https://proofwiki.org/wiki/Image_of_Idempotent_and_Directed_Suprema_Preserving_Mapping_is_Complete_Lattice | [
"Complete Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Mapping",
"Definition:Idempotence/Mapping",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Ordered Subset",
"Definition:Directed Suprema Inheriting",
"Definition:Complete Lattice"
] | [
"Definition:Directed Suprema Inheriting",
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Definition:Ordered Subset",
"Definition:Directed Subset",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Idempotence/Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Defini... |
proofwiki-12790 | Are All Perfect Numbers Even?/Progress/Minimum Size | It had been established by $1986$ that an odd perfect number, if one were to exist, would have over $200$ digits.
By $1997$ that lower bound had been raised to $300$ digits.
By $2012$ that lower bound had been raised again to $1500$ digits. | {{ProofWanted|Details}} | It had been established by $1986$ that an [[Definition:Odd Integer|odd]] [[Definition:Perfect Number|perfect number]], if one were to exist, would have over $200$ digits.
By $1997$ that lower bound had been raised to $300$ digits.
By $2012$ that lower bound had been raised again to $1500$ digits. | {{ProofWanted|Details}} | Are All Perfect Numbers Even?/Progress/Minimum Size | https://proofwiki.org/wiki/Are_All_Perfect_Numbers_Even?/Progress/Minimum_Size | https://proofwiki.org/wiki/Are_All_Perfect_Numbers_Even?/Progress/Minimum_Size | [
"Are All Perfect Numbers Even?"
] | [
"Definition:Odd Integer",
"Definition:Perfect Number"
] | [] |
proofwiki-12791 | Divisibility of Sum of 3 Fourth Powers | Let $n \in \Z_{\ge 0}$ be the sum of three $4$th powers.
Then:
:$n$ is divisible by $5$ {{iff}} all three addends are also divisible by $5$
:$n$ is divisible by $29$ {{iff}} all three addends are also divisible by $29$. | Let $n = a^4 + b^4 + c^4$ for $a, b, c \in \Z$. | Let $n \in \Z_{\ge 0}$ be the [[Definition:Integer Addition|sum]] of three [[Definition:Integer Power|$4$th powers]].
Then:
:$n$ is [[Definition:Divisor of Integer|divisible]] by $5$ {{iff}} all three [[Definition:Addend|addends]] are also [[Definition:Divisor of Integer|divisible]] by $5$
:$n$ is [[Definition:Divisor... | Let $n = a^4 + b^4 + c^4$ for $a, b, c \in \Z$. | Divisibility of Sum of 3 Fourth Powers | https://proofwiki.org/wiki/Divisibility_of_Sum_of_3_Fourth_Powers | https://proofwiki.org/wiki/Divisibility_of_Sum_of_3_Fourth_Powers | [
"Number Theory",
"29",
"5"
] | [
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Summand",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition/Summand",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-12792 | Squares of form 2 n^2 - 1 | The sequence of integers $\left\langle{n}\right\rangle$ such that $2 n^2 - 1$ is square begins:
:$1, 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, \ldots$
{{OEIS|A001653}} | {{ProofWanted|Follows somehow from the fact that these numbers are the hypotenuses of Definition:Almost Isosceles Pythagorean Triangles.}} | The [[Definition:Integer Sequence|sequence of integers]] $\left\langle{n}\right\rangle$ such that $2 n^2 - 1$ is [[Definition:Square Number|square]] begins:
:$1, 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, \ldots$
{{OEIS|A001653}} | {{ProofWanted|Follows somehow from the fact that these numbers are the hypotenuses of [[Definition:Almost Isosceles Pythagorean Triangle]]s.}} | Squares of form 2 n^2 - 1 | https://proofwiki.org/wiki/Squares_of_form_2_n^2_-_1 | https://proofwiki.org/wiki/Squares_of_form_2_n^2_-_1 | [
"Square Numbers",
"Pythagorean Triangles"
] | [
"Definition:Integer Sequence",
"Definition:Square Number"
] | [
"Definition:Almost Isosceles Pythagorean Triangle"
] |
proofwiki-12793 | Prime-Generating Quadratics of form 2 a squared plus p | The quadratic form:
:$2 a^2 + p$
yields prime numbers for $a = 0, 1, \ldots, p - 1$ for values of $p$:
:$3, 5, 11, 29$ | === 3 ===
{{:Prime-Generating Quadratics of form 2 a squared plus p/3}} | The quadratic form:
:$2 a^2 + p$
yields [[Definition:Prime Number|prime numbers]] for $a = 0, 1, \ldots, p - 1$ for values of $p$:
:$3, 5, 11, 29$ | === [[Prime-Generating Quadratics of form 2 a squared plus p/3|3]] ===
{{:Prime-Generating Quadratics of form 2 a squared plus p/3}} | Prime-Generating Quadratics of form 2 a squared plus p | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p | [
"Prime Numbers",
"Prime-Generating Quadratics of form 2 a squared plus p",
"Polynomial Expressions for Primes"
] | [
"Definition:Prime Number"
] | [
"Prime-Generating Quadratics of form 2 a squared plus p/3"
] |
proofwiki-12794 | Prime-Generating Quadratics of form 2 a squared plus p/3 | The quadratic form:
:$2 a^2 + 3$
yields prime numbers for $a = 0, 1, 2$ but not for $a = 3$. | {{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 3
| r = 0 + 3
}}
{{eqn | r = 3
| c = which is prime
}}
{{eqn | l = 2 \times 1^2 + 3
| r = 2 + 3
| c =
}}
{{eqn | r = 5
| c = which is prime
}}
{{eqn | l = 2 \times 2^2 + 3
| r = 2 \times 4 + 3
| c =
}}
{{eqn | r = 8 + 3
| c =
}}... | The quadratic form:
:$2 a^2 + 3$
yields [[Definition:Prime Number|prime numbers]] for $a = 0, 1, 2$ but not for $a = 3$. | {{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 3
| r = 0 + 3
}}
{{eqn | r = 3
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 2 \times 1^2 + 3
| r = 2 + 3
| c =
}}
{{eqn | r = 5
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 2 \times 2^2 + 3
| r = 2 \times... | Prime-Generating Quadratics of form 2 a squared plus p/3 | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p/3 | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p/3 | [
"Prime-Generating Quadratics of form 2 a squared plus p",
"3"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-12795 | Prime-Generating Quadratics of form 2 a squared plus p/5 | The quadratic form:
:$2 a^2 + 5$
yields prime numbers for $a = 0, 1, \ldots, 4$. | {{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 5
| r = 0 + 5
}}
{{eqn | r = 5
| c = which is prime
}}
{{eqn | l = 2 \times 1^2 + 5
| r = 2 + 5
| c =
}}
{{eqn | r = 7
| c = which is prime
}}
{{eqn | l = 2 \times 2^2 + 5
| r = 2 \times 4 + 5
| c =
}}
{{eqn | r = 8 + 5
| c =
}}... | The quadratic form:
:$2 a^2 + 5$
yields [[Definition:Prime Number|prime numbers]] for $a = 0, 1, \ldots, 4$. | {{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 5
| r = 0 + 5
}}
{{eqn | r = 5
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 2 \times 1^2 + 5
| r = 2 + 5
| c =
}}
{{eqn | r = 7
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 2 \times 2^2 + 5
| r = 2 \times... | Prime-Generating Quadratics of form 2 a squared plus p/5 | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p/5 | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p/5 | [
"Prime-Generating Quadratics of form 2 a squared plus p",
"5"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-12796 | Prime-Generating Quadratics of form 2 a squared plus p/11 | The quadratic form:
:$2 a^2 + 11$
yields prime numbers for $a = 0, 1, \ldots, 10$. | {{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 11
| r = 0 + 11
}}
{{eqn | r = 11
| c = which is prime
}}
{{eqn | l = 2 \times 1^2 + 11
| r = 2 + 11
| c =
}}
{{eqn | r = 13
| c = which is prime
}}
{{eqn | l = 2 \times 2^2 + 11
| r = 2 \times 4 + 11
| c =
}}
{{eqn | r = 8 + 11
... | The quadratic form:
:$2 a^2 + 11$
yields [[Definition:Prime Number|prime numbers]] for $a = 0, 1, \ldots, 10$. | {{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 11
| r = 0 + 11
}}
{{eqn | r = 11
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 2 \times 1^2 + 11
| r = 2 + 11
| c =
}}
{{eqn | r = 13
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 2 \times 2^2 + 11
| r = 2... | Prime-Generating Quadratics of form 2 a squared plus p/11 | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p/11 | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p/11 | [
"Prime-Generating Quadratics of form 2 a squared plus p",
"11"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
... |
proofwiki-12797 | Prime-Generating Quadratics of form 2 a squared plus p/29 | The quadratic form:
:$2 a^2 + 29$
yields prime numbers for $a = 0, 1, \ldots, 28$. | {{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 29
| r = 0 + 29
}}
{{eqn | r = 29
| c = which is prime
}}
{{eqn | l = 2 \times 1^2 + 29
| r = 2 + 29
| c =
}}
{{eqn | r = 31
| c = which is prime
}}
{{eqn | l = 2 \times 2^2 + 29
| r = 2 \times 4 + 29
| c =
}}
{{eqn | r = 8 + 29
... | The quadratic form:
:$2 a^2 + 29$
yields [[Definition:Prime Number|prime numbers]] for $a = 0, 1, \ldots, 28$. | {{begin-eqn}}
{{eqn | l = 2 \times 0^2 + 29
| r = 0 + 29
}}
{{eqn | r = 29
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 2 \times 1^2 + 29
| r = 2 + 29
| c =
}}
{{eqn | r = 31
| c = which is [[Definition:Prime Number|prime]]
}}
{{eqn | l = 2 \times 2^2 + 29
| r = 2... | Prime-Generating Quadratics of form 2 a squared plus p/29 | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p/29 | https://proofwiki.org/wiki/Prime-Generating_Quadratics_of_form_2_a_squared_plus_p/29 | [
"Prime-Generating Quadratics of form 2 a squared plus p",
"29"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",... |
proofwiki-12798 | Directed Suprema Preserving Mapping is Increasing | Let $L = \struct {S, \vee, \preceq}$ be a join semilattice.
Let $f: S \to S$ be a mapping that preserves directed suprema.
Then $f$ is an increasing mapping. | Let $x, y \in D$ such that
:$x \preceq y$
Then by definition of reflexivity:
:$\forall a, b \in \set {x, y}: \exists z \in \set {x, y}: a \preceq z \land b \preceq z$
By definition:
:$\set {x, y}$ is directed.
By definition of mapping preserves directed suprema:
:$f$ preserves the supremum of $\set {x, y}$.
By definiti... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $f: S \to S$ be a [[Definition:Mapping|mapping]] that [[Definition:Mapping Preserves Supremum/Directed|preserves directed suprema]].
Then $f$ is an [[Definition:Increasing Mapping|increasing mapping]]. | Let $x, y \in D$ such that
:$x \preceq y$
Then by definition of [[Definition:Reflexivity|reflexivity]]:
:$\forall a, b \in \set {x, y}: \exists z \in \set {x, y}: a \preceq z \land b \preceq z$
By definition:
:$\set {x, y}$ is [[Definition:Directed Subset|directed]].
By definition of [[Definition:Mapping Preserves S... | Directed Suprema Preserving Mapping is Increasing | https://proofwiki.org/wiki/Directed_Suprema_Preserving_Mapping_is_Increasing | https://proofwiki.org/wiki/Directed_Suprema_Preserving_Mapping_is_Increasing | [
"Increasing Mappings"
] | [
"Definition:Join Semilattice",
"Definition:Mapping",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Increasing/Mapping"
] | [
"Definition:Reflexivity",
"Definition:Directed Subset",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Mapping Preserves Supremum/Subset",
"Definition:Join Semilattice",
"Definition:Supremum of Set",
"Preceding iff Join equals Larger Operand",
"Image of Doubleton under Mapping",
"Defi... |
proofwiki-12799 | Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 1 | Let the function $\map h x$ satisfy the equation:
:$-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$
Let $\map h x$ have the boundary conditions:
:$\map h a = \map h b = 0$
Then:
:$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x = 0$ | {{begin-eqn}}
{{eqn | l = 0
| r = \int_a^b \paren 0 h \rd x
}}
{{eqn | r = \int_a^b \paren {-\map {\frac \d {\d x} } {P h'} + Q h} h \rd x
}}
{{eqn | r = \int_a^b Q h^2 \rd x - \int_a^b \map {\frac \d {\d x} } {P h'} h \rd x
}}
{{eqn | r = \int_a^b Q h^2 \rd x - \bigintlimits {P h' h} a b + \int_a^b P h' \rd h
... | Let the function $\map h x$ satisfy the equation:
:$-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$
Let $\map h x$ have the boundary conditions:
:$\map h a = \map h b = 0$
Then:
:$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x = 0$ | {{begin-eqn}}
{{eqn | l = 0
| r = \int_a^b \paren 0 h \rd x
}}
{{eqn | r = \int_a^b \paren {-\map {\frac \d {\d x} } {P h'} + Q h} h \rd x
}}
{{eqn | r = \int_a^b Q h^2 \rd x - \int_a^b \map {\frac \d {\d x} } {P h'} h \rd x
}}
{{eqn | r = \int_a^b Q h^2 \rd x - \bigintlimits {P h' h} a b + \int_a^b P h' \rd h
... | Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 1 | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Quadratic_Functional_to_be_Positive_Definite/Lemma_1 | https://proofwiki.org/wiki/Necessary_and_Sufficient_Condition_for_Quadratic_Functional_to_be_Positive_Definite/Lemma_1 | [
"Calculus of Variations"
] | [] | [
"Integration by Parts"
] |
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