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-13200 | Numbers not Expressible as Sum of Distinct Pentagonal Numbers | The positive integers which cannot be expressed as the sum of distinct pentagonal numbers are:
:$2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30,$
:$31, 32, 33, 37, 38, 42, 43, 44, 45, 46, 49, 50, 54, 55, 59, 60, 61, 65,$
:$66, 67, 72, 77, 80, 81, 84, 89, 94, 95, 96, 100, 101, 102, 107, 112, 116,$
... | It will be proved that the largest integer which cannot be expressed as the sum of distinct pentagonal numbers is $159$.
The remaining integers in the sequence can be identified by inspection.
We prove this using a variant of Second Principle of Mathematical Induction.
Let $\map P n$ be the proposition:
:$n$ can be exp... | The [[Definition:Positive Integer|positive integers]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]] are:
:$2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30,$
:$31, 32, 33, 37, 38, 42, 43, ... | It will be proved that the largest [[Definition:Integer|integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct Elements|distinct]] [[Definition:Pentagonal Number|pentagonal numbers]] is $159$.
The remaining [[Definition:Integer|integers]] in the [[Definition:Integer Seq... | Numbers not Expressible as Sum of Distinct Pentagonal Numbers | https://proofwiki.org/wiki/Numbers_not_Expressible_as_Sum_of_Distinct_Pentagonal_Numbers | https://proofwiki.org/wiki/Numbers_not_Expressible_as_Sum_of_Distinct_Pentagonal_Numbers | [
"Pentagonal Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Pentagonal Number"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Pentagonal Number",
"Definition:Integer",
"Definition:Integer Sequence",
"Second Principle of Mathematical Induction",
"Definition:Addition/Integers",
"Definition:Distinct/Plural",
"Definition:Pentagona... |
proofwiki-13201 | Equivalence of Definitions of Polynomial Ring in One Variable | Let $R$ be a commutative ring with unity.
The following definitions of polynomial ring are equivalent in the following sense:
: For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates.
{{explain|this statement has to be made more precise}}
=== Definition 1: As a Ring of... | Use Polynomial Ring of Sequences Satisfies Universal Property
{{ProofWanted}} | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
The following definitions of [[Definition:Polynomial Ring|polynomial ring]] are equivalent in the following sense:
: For every two constructions, there exists an [[Definition:R-isomorphism|$R$-isomorphism]] which sends [[Definition:I... | Use [[Polynomial Ring of Sequences Satisfies Universal Property]]
{{ProofWanted}} | Equivalence of Definitions of Polynomial Ring in One Variable | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Polynomial_Ring_in_One_Variable | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Polynomial_Ring_in_One_Variable | [
"Polynomial Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:R-isomorphism",
"Definition:Indeterminate",
"Definition:Polynomial Ring/Sequences",
"Definition:Polynomial Ring/Monoid Ring on Natural Numbers"
] | [
"Polynomial Ring of Sequences Satisfies Universal Property"
] |
proofwiki-13202 | Smallest Triplet of Integers whose Product with Divisor Count are Equal | Let $\map {\sigma_0} n$ denote the divisor count function: the number of divisors of $n$.
The smallest set of $3$ integers $T$ such that $m \, \map {\sigma_0} m$ is equal for each $m \in T$ is:
:$\set {168, 192, 224}$ | {{begin-eqn}}
{{eqn | l = 168 \times \map {\sigma_0} {168}
| r = 168 \times 16
| c = {{DCFLink|168}}
}}
{{eqn | r = \paren {2^3 \times 3 \times 7} \times 2^4
| c =
}}
{{eqn | r = 2^7 \times 3 \times 7
| c =
}}
{{eqn | r = 2688
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 192 \times \m... | Let $\map {\sigma_0} n$ denote the [[Definition:Divisor Count Function|divisor count function]]: the number of [[Definition:Divisor of Integer|divisors]] of $n$.
The smallest [[Definition:Set|set]] of $3$ [[Definition:Positive Integer|integers]] $T$ such that $m \, \map {\sigma_0} m$ is equal for each $m \in T$ is:
:$... | {{begin-eqn}}
{{eqn | l = 168 \times \map {\sigma_0} {168}
| r = 168 \times 16
| c = {{DCFLink|168}}
}}
{{eqn | r = \paren {2^3 \times 3 \times 7} \times 2^4
| c =
}}
{{eqn | r = 2^7 \times 3 \times 7
| c =
}}
{{eqn | r = 2688
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 192 \times ... | Smallest Triplet of Integers whose Product with Divisor Count are Equal | https://proofwiki.org/wiki/Smallest_Triplet_of_Integers_whose_Product_with_Divisor_Count_are_Equal | https://proofwiki.org/wiki/Smallest_Triplet_of_Integers_whose_Product_with_Divisor_Count_are_Equal | [
"Divisor Count Function",
"168",
"192",
"224"
] | [
"Definition:Divisor Count Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Set",
"Definition:Positive/Integer"
] | [] |
proofwiki-13203 | General Variation of Integral Functional/Dependent on n Variables | Let $\mathbf x$ be an $n$-dimensional vector.
Let $u = \map u {\mathbf x}$ be a real-valued function.
Let $J$ be a functional such that:
:$\ds J \sqbrk u = \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 \dotsm \rd x_n$
Let $\mathbf x^*, u^*$ be defined by the following transformations $... | By definition:
{{begin-eqn}}
{{eqn | l = \Delta J
| r = J \sqbrk {\map {u^*} {x^*} } - J \sqbrk {\map u x}
}}
{{eqn | r = \int_{R^*} \map F {\mathbf x^*, u^*, \frac {\partial u^*} {\partial \mathbf x^*} } \rd x_1^* \dotsm \rd x_n^* - \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1 ... | Let $\mathbf x$ be an [[Definition:Dimension|$n$-dimensional]] [[Definition:Vector|vector]].
Let $u = \map u {\mathbf x}$ be a [[Definition:Real-Valued Function|real-valued function]].
Let $J$ be a [[Definition:Real Functional|functional]] such that:
:$\ds J \sqbrk u = \int_R \map F {\mathbf x, u, \dfrac {\partial u... | By definition:
{{begin-eqn}}
{{eqn | l = \Delta J
| r = J \sqbrk {\map {u^*} {x^*} } - J \sqbrk {\map u x}
}}
{{eqn | r = \int_{R^*} \map F {\mathbf x^*, u^*, \frac {\partial u^*} {\partial \mathbf x^*} } \rd x_1^* \dotsm \rd x_n^* - \int_R \map F {\mathbf x, u, \dfrac {\partial u} {\partial \mathbf x} } \rd x_1... | General Variation of Integral Functional/Dependent on n Variables | https://proofwiki.org/wiki/General_Variation_of_Integral_Functional/Dependent_on_n_Variables | https://proofwiki.org/wiki/General_Variation_of_Integral_Functional/Dependent_on_n_Variables | [
"Calculus of Variations"
] | [
"Definition:Dimension",
"Definition:Vector",
"Definition:Real-Valued Function",
"Definition:Functional/Real",
"Definition:Differentiable Mapping",
"Definition:Differential of Mapping/Functional",
"Definition:Functional/Real",
"Definition:Mapping"
] | [
"Definition:Jacobian/Matrix",
"Taylor's Theorem",
"Definition:Jacobian/Determinant",
"Definition:Diagonal",
"Definition:Multiplication/Product",
"Definition:Diagonal",
"Definition:Multiplication/Product",
"Definition:Differentiable Functional",
"Definition:Difference",
"Definition:Difference",
"... |
proofwiki-13204 | Pairs of Integers whose Product with Divisor Count are Equal | Let $\map {\sigma_0} n$ denote the divisor count function: the number of divisors of $n$.
The following pairs of integers $T$ have the property that $m \, \map {\sigma_0} m$ is equal for each $m \in T$:
:$\set {18, 27}$
:$\set {24, 32}$
:$\set {56, 64}$ | {{begin-eqn}}
{{eqn | l = 18 \times \map {\sigma_0} {18}
| r = 18 \times 6
| c = {{DCFLink|18}}
}}
{{eqn | r = \paren {2 \times 3^2} \times \paren {2 \times 3}
| c =
}}
{{eqn | r = 2^2 \times 3^3
| c =
}}
{{eqn | r = 108
| c =
}}
{{eqn | l = 27 \times \map {\sigma_0} {27}
| r = 27... | Let $\map {\sigma_0} n$ denote the [[Definition:Divisor Count Function|divisor count function]]: the number of [[Definition:Divisor of Integer|divisors]] of $n$.
The following [[Definition:Doubleton|pairs]] of [[Definition:Positive Integer|integers]] $T$ have the property that $m \, \map {\sigma_0} m$ is equal for eac... | {{begin-eqn}}
{{eqn | l = 18 \times \map {\sigma_0} {18}
| r = 18 \times 6
| c = {{DCFLink|18}}
}}
{{eqn | r = \paren {2 \times 3^2} \times \paren {2 \times 3}
| c =
}}
{{eqn | r = 2^2 \times 3^3
| c =
}}
{{eqn | r = 108
| c =
}}
{{eqn | l = 27 \times \map {\sigma_0} {27}
| r = 27... | Pairs of Integers whose Product with Divisor Count are Equal | https://proofwiki.org/wiki/Pairs_of_Integers_whose_Product_with_Divisor_Count_are_Equal | https://proofwiki.org/wiki/Pairs_of_Integers_whose_Product_with_Divisor_Count_are_Equal | [
"Divisor Count Function"
] | [
"Definition:Divisor Count Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Doubleton",
"Definition:Positive/Integer"
] | [] |
proofwiki-13205 | Order Isomorphism forms Galois Connection | Let $L_1 = \struct {S_1, \preceq_1}$, $L_2 = \struct {S_2, \preceq_2}$ be ordered sets.
Let $f:S_1 \to S_2$ be an order isomorphism between $L_1$ and $L_2$.
Then $\struct {f, f^{-1} }$ is a Galois connection. | Let $t \in S_2$, $s \in S_1$.
We will prove that:
:$t \preceq_2 \map f s \implies \map {f^{-1} } t \preceq_1 s$
Assume that:
:$t \preceq_2 \map f s$
By Inverse of Order Isomorphism is Order Isomorphism:
:$f^{-1}$ is an order isomorphism.
By definition of order isomorphism:
:$f^{-1}$ is an order embedding.
By definition... | Let $L_1 = \struct {S_1, \preceq_1}$, $L_2 = \struct {S_2, \preceq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $f:S_1 \to S_2$ be an [[Definition:Order Isomorphism|order isomorphism]] between $L_1$ and $L_2$.
Then $\struct {f, f^{-1} }$ is a [[Definition:Galois Connection|Galois connection]]. | Let $t \in S_2$, $s \in S_1$.
We will prove that:
:$t \preceq_2 \map f s \implies \map {f^{-1} } t \preceq_1 s$
Assume that:
:$t \preceq_2 \map f s$
By [[Inverse of Order Isomorphism is Order Isomorphism]]:
:$f^{-1}$ is an [[Definition:Order Isomorphism|order isomorphism]].
By definition of [[Definition:Order Isomo... | Order Isomorphism forms Galois Connection | https://proofwiki.org/wiki/Order_Isomorphism_forms_Galois_Connection | https://proofwiki.org/wiki/Order_Isomorphism_forms_Galois_Connection | [
"Galois Connections",
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Order Isomorphism",
"Definition:Galois Connection"
] | [
"Inverse of Order Isomorphism is Order Isomorphism",
"Definition:Order Isomorphism",
"Definition:Order Isomorphism",
"Definition:Order Embedding",
"Definition:Order Embedding",
"Definition:Bijection",
"Definition:Order Embedding",
"Definition:Bijection"
] |
proofwiki-13206 | Sequence of Square Centered Hexagonal Numbers | The sequence of centered hexagonal numbers which are also square begins:
:$1, 169, 32 \, 761, 6 \, 355 \, 441, \ldots$
{{OEIS|A006051}} | {{:Definition:Centered Hexagonal Number/Sequence}}
We have that:
{{begin-eqn}}
{{eqn | l = 1
| r = 1^2
| c = {{Defof|Square Number}}
}}
{{eqn | r = 3 \times 1 \times \paren {1 - 1} + 1
| c = Closed Form for Centered Hexagonal Numbers
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 169
| r = 13^2
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Centered Hexagonal Number|centered hexagonal numbers]] which are also [[Definition:Square Number|square]] begins:
:$1, 169, 32 \, 761, 6 \, 355 \, 441, \ldots$
{{OEIS|A006051}} | {{:Definition:Centered Hexagonal Number/Sequence}}
We have that:
{{begin-eqn}}
{{eqn | l = 1
| r = 1^2
| c = {{Defof|Square Number}}
}}
{{eqn | r = 3 \times 1 \times \paren {1 - 1} + 1
| c = [[Closed Form for Centered Hexagonal Numbers]]
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 169
| r = 13^2... | Sequence of Square Centered Hexagonal Numbers | https://proofwiki.org/wiki/Sequence_of_Square_Centered_Hexagonal_Numbers | https://proofwiki.org/wiki/Sequence_of_Square_Centered_Hexagonal_Numbers | [
"Centered Hexagonal Numbers",
"Square Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Centered Hexagonal Number",
"Definition:Square Number"
] | [
"Closed Form for Centered Hexagonal Numbers",
"Closed Form for Centered Hexagonal Numbers",
"Closed Form for Centered Hexagonal Numbers",
"Closed Form for Centered Hexagonal Numbers"
] |
proofwiki-13207 | 169 as Sum of up to 155 Squares | $169$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $1$ to $155$. | We note the following:
{{begin-eqn}}
{{eqn | l = 169
| r = 13^2
| c =
}}
{{eqn | r = 12^2 + 5^2
| c =
}}
{{eqn | r = 12^2 + 4^2 + 3^2
| c =
}}
{{eqn | r = 8^2 + 8^2 + 5^2 + 4^2
| c =
}}
{{eqn | r = 8^2 + 8^2 + 4^2 + 4^2 + 3^2
| c =
}}
{{end-eqn}}
Let $41 \le n \le 155$.
Let $n \... | $169$ can be expressed as the [[Definition:Integer Addition|sum]] of $n$ non-zero [[Definition:Square Number|squares]] for all $n$ from $1$ to $155$. | We note the following:
{{begin-eqn}}
{{eqn | l = 169
| r = 13^2
| c =
}}
{{eqn | r = 12^2 + 5^2
| c =
}}
{{eqn | r = 12^2 + 4^2 + 3^2
| c =
}}
{{eqn | r = 8^2 + 8^2 + 5^2 + 4^2
| c =
}}
{{eqn | r = 8^2 + 8^2 + 4^2 + 4^2 + 3^2
| c =
}}
{{end-eqn}}
Let $41 \le n \le 155$.
Let ... | 169 as Sum of up to 155 Squares | https://proofwiki.org/wiki/169_as_Sum_of_up_to_155_Squares | https://proofwiki.org/wiki/169_as_Sum_of_up_to_155_Squares | [
"Sums of Squares",
"169"
] | [
"Definition:Addition/Integers",
"Definition:Square Number"
] | [
"Definition:Square Number",
"25 as Sum of 4 to 11 Squares",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Square Number"... |
proofwiki-13208 | Lattice of Power Set is Arithmetic | Let $X$ be a set.
Let $P = \struct {\powerset X, \cup, \cap, \subseteq}$ be a lattice of power set.
Then $P$ is an arithmetic ordered set. | Define $C = \struct {\map K P, \preceq}$ as an ordered subset of $P$
where $\map K P$ denotes the compact subset of $P$.
Thus by Lattice of Power Set is Algebraic:
:$P$ is algebraic.
It remains to prove that:
:$\map K P$ is meet closed.
Let $x, y \in \map K P$.
By definition of compact subset:
:$x$ is compact.
By Eleme... | Let $X$ be a [[Definition:Set|set]].
Let $P = \struct {\powerset X, \cup, \cap, \subseteq}$ be a [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]].
Then $P$ is an [[Definition:Arithmetic Ordered Set|arithmetic ordered set]]. | Define $C = \struct {\map K P, \preceq}$ as an [[Definition:Ordered Subset|ordered subset]] of $P$
where $\map K P$ denotes the [[Definition:Compact Subset of Lattice|compact subset]] of $P$.
Thus by [[Lattice of Power Set is Algebraic]]:
:$P$ is [[Definition:Algebraic Ordered Set|algebraic]].
It remains to prove th... | Lattice of Power Set is Arithmetic | https://proofwiki.org/wiki/Lattice_of_Power_Set_is_Arithmetic | https://proofwiki.org/wiki/Lattice_of_Power_Set_is_Arithmetic | [
"Continuous Lattices"
] | [
"Definition:Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Arithmetic Ordered Set"
] | [
"Definition:Ordered Subset",
"Definition:Compact Subset of Lattice",
"Lattice of Power Set is Algebraic",
"Definition:Algebraic Ordered Set",
"Definition:Meet Closed",
"Definition:Compact Subset of Lattice",
"Definition:Compact Element",
"Element is Finite iff Element is Compact in Lattice of Power Se... |
proofwiki-13209 | Nth Derivative of Natural Logarithm | The $n$th derivative of $\map \ln x$ for $n \ge 1$ is:
:$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$ | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$ | The [[Definition:Higher Derivative|$n$th derivative]] of $\map \ln x$ for $n \ge 1$ is:
:$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$ | Nth Derivative of Natural Logarithm | https://proofwiki.org/wiki/Nth_Derivative_of_Natural_Logarithm | https://proofwiki.org/wiki/Nth_Derivative_of_Natural_Logarithm | [
"Derivative of Natural Logarithm Function",
"Natural Logarithms",
"Proofs by Induction"
] | [
"Definition:Derivative/Higher Derivatives/Higher Order"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-13210 | Sum of Factorials of Digits of 169 | Let the factorials of the digits of $169$ be added.
Let the same process be done on the result.
Repeat.
After $3$ iterations, the result will be $169$. | {{begin-eqn}}
{{eqn | l = 1! + 6! + 9!
| r = 1 + 720 + 362 \, 880
| c =
}}
{{eqn | r = 363 \, 601
| c =
}}
{{eqn | l = 3! + 6! + 3! + 6! + 0! + 1!
| r = 6 + 720 + 6 + 720 + 1 + 1
| c =
}}
{{eqn | r = 1454
| c =
}}
{{eqn | l = 1! + 4! + 5! + 4!
| r = 1 + 24 + 120 + 24
... | Let the [[Definition:Factorial|factorials]] of the [[Definition:Digit|digits]] of $169$ be [[Definition:Integer Addition|added]].
Let the same process be done on the result.
Repeat.
After $3$ iterations, the result will be $169$. | {{begin-eqn}}
{{eqn | l = 1! + 6! + 9!
| r = 1 + 720 + 362 \, 880
| c =
}}
{{eqn | r = 363 \, 601
| c =
}}
{{eqn | l = 3! + 6! + 3! + 6! + 0! + 1!
| r = 6 + 720 + 6 + 720 + 1 + 1
| c =
}}
{{eqn | r = 1454
| c =
}}
{{eqn | l = 1! + 4! + 5! + 4!
| r = 1 + 24 + 120 + 24
... | Sum of Factorials of Digits of 169 | https://proofwiki.org/wiki/Sum_of_Factorials_of_Digits_of_169 | https://proofwiki.org/wiki/Sum_of_Factorials_of_Digits_of_169 | [
"169",
"Factorials"
] | [
"Definition:Factorial",
"Definition:Digit",
"Definition:Addition/Integers"
] | [] |
proofwiki-13211 | Squares which are Difference between Two Cubes | $169$ is the smallest square number which is the difference between two cubes:
:$169 = 8^3 - 7^3$
{{expand|Add the rest of the sequence, having found out what they are.}} | {{begin-eqn}}
{{eqn | l = 8^3 - 7^3
| r = 512 - 343
| c =
}}
{{eqn | r = 169
| c =
}}
{{eqn | r = 13^2
| c =
}}
{{end-eqn}}
{{ProofWanted|Establish that this is the smallest such.}} | $169$ is the smallest [[Definition:Square Number|square number]] which is the [[Definition:Integer Subtraction|difference]] between two [[Definition:Cube Number|cubes]]:
:$169 = 8^3 - 7^3$
{{expand|Add the rest of the sequence, having found out what they are.}} | {{begin-eqn}}
{{eqn | l = 8^3 - 7^3
| r = 512 - 343
| c =
}}
{{eqn | r = 169
| c =
}}
{{eqn | r = 13^2
| c =
}}
{{end-eqn}}
{{ProofWanted|Establish that this is the smallest such.}} | Squares which are Difference between Two Cubes | https://proofwiki.org/wiki/Squares_which_are_Difference_between_Two_Cubes | https://proofwiki.org/wiki/Squares_which_are_Difference_between_Two_Cubes | [
"169",
"Square Numbers",
"Cube Numbers"
] | [
"Definition:Square Number",
"Definition:Subtraction/Integers",
"Definition:Cube Number"
] | [] |
proofwiki-13212 | Cube of 180 is Sum of Sequence of Consecutive Cubes | :$180^3 = \ds \sum_{k \mathop = 6}^{69} k^3$
That is:
:$180^3 = 6^3 + 7^3 + \cdots + 67^3 + 68^3 + 69^3$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{69} k^3
| r = \paren {\dfrac {69 \paren {69 + 1} } 2}^2
| c = Sum of Sequence of Cubes
}}
{{eqn | r = 5 \, 832 \, 225
| c =
}}
{{eqn | l = \sum_{k \mathop = 1}^5 k^3
| r = \paren {\dfrac {5 \paren {5 + 1} } 2}^2
| c = Sum of Sequence of Cube... | :$180^3 = \ds \sum_{k \mathop = 6}^{69} k^3$
That is:
:$180^3 = 6^3 + 7^3 + \cdots + 67^3 + 68^3 + 69^3$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{69} k^3
| r = \paren {\dfrac {69 \paren {69 + 1} } 2}^2
| c = [[Sum of Sequence of Cubes]]
}}
{{eqn | r = 5 \, 832 \, 225
| c =
}}
{{eqn | l = \sum_{k \mathop = 1}^5 k^3
| r = \paren {\dfrac {5 \paren {5 + 1} } 2}^2
| c = [[Sum of Sequence o... | Cube of 180 is Sum of Sequence of Consecutive Cubes | https://proofwiki.org/wiki/Cube_of_180_is_Sum_of_Sequence_of_Consecutive_Cubes | https://proofwiki.org/wiki/Cube_of_180_is_Sum_of_Sequence_of_Consecutive_Cubes | [
"Cube Numbers",
"180"
] | [] | [
"Sum of Sequence of Cubes",
"Sum of Sequence of Cubes"
] |
proofwiki-13213 | 3-Digit Numbers forming Longest Reverse-and-Add Sequence | Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation.
Let $r \left({m}\right)$ be the reverse-and-add process on $m$.
Let $r$ be applied iteratively to $m$.
The $3$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are:
:$187, 286, 385, 583, 682, 781,... | The sequence obtained by iterating $r$ on $187$ is:
:$187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
:$17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
:$47267087164, 93445163438, 176881317877, 955594506548, 170120002107, 8713200023178$
{{OEIS|A033670}}... | Let $m \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]] expressed in [[Definition:Decimal Notation|decimal notation]].
Let $r \left({m}\right)$ be the [[Definition:Reverse-and-Add|reverse-and-add process]] on $m$.
Let $r$ be applied iteratively to $m$.
The $3$-[[Definition:Digit|digit]] [[Definiti... | The [[Definition:Integer Sequence|sequence]] obtained by iterating $r$ on $187$ is:
:$187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
:$17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
:$47267087164, 93445163438, 176881317877, 955594506548, 170120002107,... | 3-Digit Numbers forming Longest Reverse-and-Add Sequence | https://proofwiki.org/wiki/3-Digit_Numbers_forming_Longest_Reverse-and-Add_Sequence | https://proofwiki.org/wiki/3-Digit_Numbers_forming_Longest_Reverse-and-Add_Sequence | [
"Reverse-and-Add"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Notation",
"Definition:Reverse-and-Add",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Palindromic Number",
"Definition:Palindromic Number"
] | [
"Definition:Integer Sequence",
"Definition:Integer Sequence"
] |
proofwiki-13214 | Numbers Expressible as Sum of Five Distinct Squares | The largest positive integer which cannot be expressed as the sum of no more than $5$ distinct squares is $188$.
Both $188$ and $124$ require as many as $6$ distinct squares to represent them:
{{begin-eqn}}
{{eqn | l = 124
| r = 1 + 4 + 9 + 25 + 36 + 49
| c =
}}
{{eqn | r = 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 7^... | From Numbers not Sum of Distinct Squares, the following positive integers cannot be expressed as the sum of distinct squares at all:
:$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128$
{{OEIS|A001422}}
{{ProofWanted|It remains to be shown that all ... | The largest [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of no more than $5$ [[Definition:Distinct|distinct]] [[Definition:Square Number|squares]] is $188$.
Both $188$ and $124$ require as many as $6$ [[Definition:Distinct|distinct]] [[Definitio... | From [[Numbers not Sum of Distinct Squares]], the following [[Definition:Positive Integer|positive integers]] cannot be expressed as the [[Definition:Integer Addition|sum]] of [[Definition:Distinct|distinct]] [[Definition:Square Number|squares]] at all:
:$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 3... | Numbers Expressible as Sum of Five Distinct Squares | https://proofwiki.org/wiki/Numbers_Expressible_as_Sum_of_Five_Distinct_Squares | https://proofwiki.org/wiki/Numbers_Expressible_as_Sum_of_Five_Distinct_Squares | [
"Specific Numbers",
"188"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Square Number",
"Definition:Distinct",
"Definition:Square Number"
] | [
"Numbers not Sum of Distinct Squares",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Square Number",
"Definition:Positive/Integer"
] |
proofwiki-13215 | Pandigital Sum whose Components are Multiples | The following sums use each of the digits $1$ to $9$ exactly once each, while one summand equals $2$ times the other summand:
{{begin-eqn}}
{{eqn | l = 192 + 384
| r = 576
| c = where $2 \times 192 = 384$ and so $3 \times 192 = 576$
}}
{{eqn | l = 219 + 438
| r = 657
| c = where $2 \times 219 = ... | We establish some parameters for the number $x$ such that the equation $x + 2 x = 3 x$ is pandigital.
Firstly, among the three numbers, there cannot be repeating digits or $0$.
If $5 \divides x$:
:$10 \divides 2 x$ and $2 x$ ends in $0$.
By Divisibility by 5, $x$ cannot end in $5$.
Secondly, $x$ must be divisible by $3... | The following [[Definition:Integer Addition|sums]] use each of the [[Definition:Digit|digits]] $1$ to $9$ exactly once each, while one [[Definition:Summand|summand]] equals $2$ times the other [[Definition:Summand|summand]]:
{{begin-eqn}}
{{eqn | l = 192 + 384
| r = 576
| c = where $2 \times 192 = 384$ and... | We establish some parameters for the number $x$ such that the equation $x + 2 x = 3 x$ is pandigital.
Firstly, among the three numbers, there cannot be repeating [[Definition:Digit|digits]] or $0$.
If $5 \divides x$:
:$10 \divides 2 x$ and $2 x$ ends in $0$.
By [[Divisibility by 5]], $x$ cannot end in $5$.
Secondl... | Pandigital Sum whose Components are Multiples | https://proofwiki.org/wiki/Pandigital_Sum_whose_Components_are_Multiples | https://proofwiki.org/wiki/Pandigital_Sum_whose_Components_are_Multiples | [
"Recreational Mathematics"
] | [
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Addition/Summand",
"Definition:Addition/Summand",
"Definition:Addition/Integers",
"Definition:Anagram"
] | [
"Definition:Digit",
"Divisibility by 5",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Addition/Integers",
"Divisibility by 9",
"Congruence of Powers",
"Euclid's Lemma",
"Definition:Digit"
] |
proofwiki-13216 | Closed Form for Heptagonal Pyramidal Numbers | The closed-form expression for the $n$th heptagonal pyramidal number is:
:$Q_n = \dfrac {n \paren {n + 1} \paren {5 n - 2} } 6$ | {{begin-eqn}}
{{eqn | l = Q_n
| r = \sum_{k \mathop = 1}^n H_n
| c = {{Defof|Heptagonal Pyramidal Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \dfrac {k \paren {5 k - 3} } 2
| c = Closed Form for Heptagonal Numbers
}}
{{eqn | r = \dfrac 1 2 \paren {5 \sum_{k \mathop = 1}^n k^2 - 3 \sum_{k \mathop = ... | The [[Definition:Closed-Form Expression|closed-form expression]] for the $n$th [[Definition:Heptagonal Pyramidal Number|heptagonal pyramidal number]] is:
:$Q_n = \dfrac {n \paren {n + 1} \paren {5 n - 2} } 6$ | {{begin-eqn}}
{{eqn | l = Q_n
| r = \sum_{k \mathop = 1}^n H_n
| c = {{Defof|Heptagonal Pyramidal Number}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \dfrac {k \paren {5 k - 3} } 2
| c = [[Closed Form for Heptagonal Numbers]]
}}
{{eqn | r = \dfrac 1 2 \paren {5 \sum_{k \mathop = 1}^n k^2 - 3 \sum_{k \matho... | Closed Form for Heptagonal Pyramidal Numbers | https://proofwiki.org/wiki/Closed_Form_for_Heptagonal_Pyramidal_Numbers | https://proofwiki.org/wiki/Closed_Form_for_Heptagonal_Pyramidal_Numbers | [
"Closed Forms",
"Pyramidal Numbers"
] | [
"Definition:Closed Form Expression",
"Definition:Heptagonal Pyramidal Number"
] | [
"Closed Form for Heptagonal Numbers",
"Sum of Sequence of Squares",
"Closed Form for Triangular Numbers"
] |
proofwiki-13217 | Heptagonal Pyramidal Numbers which are Square | The sequence of heptagonal pyramidal numbers which also have the property of being square begins:
:$0, 1, 196, 99 \, 225$
{{expand|Probably no more, related to elliptic curves}} | {{begin-eqn}}
{{eqn | r = \dfrac {0 \paren {0 + 1} \paren {5 \times 0 - 2} } 6
| o =
| c = Closed Form for Heptagonal Pyramidal Numbers
}}
{{eqn | r = \dfrac {0 \times 1 \times \paren {- 2} } 6
| c =
}}
{{eqn | r = 0
| c =
}}
{{eqn | r = 0^2
| c = {{Defof|Square Number}}
}}
{{end-eqn}}
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Heptagonal Pyramidal Number|heptagonal pyramidal numbers]] which also have the property of being [[Definition:Square Number|square]] begins:
:$0, 1, 196, 99 \, 225$
{{expand|Probably no more, related to elliptic curves}} | {{begin-eqn}}
{{eqn | r = \dfrac {0 \paren {0 + 1} \paren {5 \times 0 - 2} } 6
| o =
| c = [[Closed Form for Heptagonal Pyramidal Numbers]]
}}
{{eqn | r = \dfrac {0 \times 1 \times \paren {- 2} } 6
| c =
}}
{{eqn | r = 0
| c =
}}
{{eqn | r = 0^2
| c = {{Defof|Square Number}}
}}
{{end-eq... | Heptagonal Pyramidal Numbers which are Square | https://proofwiki.org/wiki/Heptagonal_Pyramidal_Numbers_which_are_Square | https://proofwiki.org/wiki/Heptagonal_Pyramidal_Numbers_which_are_Square | [
"Pyramidal Numbers",
"Square Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Heptagonal Pyramidal Number",
"Definition:Square Number"
] | [
"Closed Form for Heptagonal Pyramidal Numbers",
"Closed Form for Heptagonal Pyramidal Numbers",
"Closed Form for Heptagonal Pyramidal Numbers",
"Closed Form for Heptagonal Pyramidal Numbers"
] |
proofwiki-13218 | Conditions for Limit Function to be Limit Minimizing Function of Functional | Let $y$ be a real function.
Let $J \sqbrk y$ be a functional.
Let $\sequence {y_n}$ be a minimizing sequence of $J$.
Let:
:$\ds \lim_{n \mathop \to \infty} y_n = \hat y$
Suppose $J$ is lower semicontinuous at $y = \hat y$.
Then:
:$\ds J \sqbrk {\hat y} = \lim_{n \mathop \to \infty} J \sqbrk {y_n}$ | By definition of minimizing sequence:
:$\ds \inf_y J \sqbrk y = \lim_{n \mathop \to \infty} J \sqbrk {y_n}$
Any mapping from this sequence either minimises the functional or not.
This is true for the limit mapping as well:
:$\ds J \sqbrk {\hat y} \ge \inf_y J \sqbrk y$
By assumption, $J$ is lower semicontinuous at $\ha... | Let $y$ be a [[Definition:Real Function|real function]].
Let $J \sqbrk y$ be a [[Definition:Real Functional|functional]].
Let $\sequence {y_n}$ be a [[Definition:Minimizing Sequence of Functional|minimizing sequence]] of $J$.
Let:
:$\ds \lim_{n \mathop \to \infty} y_n = \hat y$
Suppose $J$ is [[Definition:Lower Se... | By definition of [[Definition:Minimizing Sequence of Functional|minimizing sequence]]:
:$\ds \inf_y J \sqbrk y = \lim_{n \mathop \to \infty} J \sqbrk {y_n}$
Any [[Definition:Mapping|mapping]] from this [[Definition:Sequence|sequence]] either minimises the [[Definition:Real Functional|functional]] or not.
This is tru... | Conditions for Limit Function to be Limit Minimizing Function of Functional | https://proofwiki.org/wiki/Conditions_for_Limit_Function_to_be_Limit_Minimizing_Function_of_Functional | https://proofwiki.org/wiki/Conditions_for_Limit_Function_to_be_Limit_Minimizing_Function_of_Functional | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Functional/Real",
"Definition:Sequence/Minimizing/Functional",
"Definition:Lower Semicontinuous"
] | [
"Definition:Sequence/Minimizing/Functional",
"Definition:Mapping",
"Definition:Sequence",
"Definition:Functional/Real",
"Definition:Mapping",
"Definition:Assumption",
"Definition:Lower Semicontinuous",
"Definition:Sufficiently Large",
"Definition:Sequence/Minimizing/Functional",
"Definition:Suffic... |
proofwiki-13219 | Smallest 10 Primes in Arithmetic Sequence | The smallest $10$ primes in arithmetic sequence are:
:$199 + 210 n$
for $n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$.
These are also the smallest $8$ and $9$ primes in arithmetic sequence. | {{begin-eqn}}
{{eqn | l = 199 + 0 \times 210
| r = 199
| c = which is the $46$th prime
}}
{{eqn | l = 199 + 1 \times 210
| r = 409
| c = which is the $80$th prime
}}
{{eqn | l = 199 + 2 \times 210
| r = 619
| c = which is the $114$th prime
}}
{{eqn | l = 199 + 3 \times 210
| r ... | The smallest $10$ [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] are:
:$199 + 210 n$
for $n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$.
These are also the smallest $8$ and $9$ [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]]. | {{begin-eqn}}
{{eqn | l = 199 + 0 \times 210
| r = 199
| c = which is the $46$th [[Definition:Prime Number|prime]]
}}
{{eqn | l = 199 + 1 \times 210
| r = 409
| c = which is the $80$th [[Definition:Prime Number|prime]]
}}
{{eqn | l = 199 + 2 \times 210
| r = 619
| c = which is the $1... | Smallest 10 Primes in Arithmetic Sequence | https://proofwiki.org/wiki/Smallest_10_Primes_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Smallest_10_Primes_in_Arithmetic_Sequence | [
"Prime Numbers",
"Arithmetic Sequences"
] | [
"Definition:Prime Number",
"Definition:Arithmetic Sequence",
"Definition:Prime Number",
"Definition:Arithmetic Sequence"
] | [
"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-13220 | Numbers that cannot be made Prime by changing 1 Digit | The following positive integers cannot be made into prime numbers by changing just one digit:
:$200, 202, 204, 205, 206, 208, \ldots$
{{OEIS|A192545}} | In order to make any one of these positive integers into a prime number one would have to change the last digit.
Otherwise the number it was changed into would be either even, or divisible by $5$, and so not prime.
But all the other integers between $200$ and $209$ are composite:
{{begin-eqn}}
{{eqn | l = 201
| r... | The following [[Definition:Positive Integer|positive integers]] cannot be made into [[Definition:Prime Number|prime numbers]] by changing just one [[Definition:Digit|digit]]:
:$200, 202, 204, 205, 206, 208, \ldots$
{{OEIS|A192545}} | In order to make any one of these [[Definition:Positive Integer|positive integers]] into a [[Definition:Prime Number|prime number]] one would have to change the last [[Definition:Digit|digit]].
Otherwise the number it was changed into would be either [[Definition:Even Integer|even]], or [[Definition:Divisor of Integer... | Numbers that cannot be made Prime by changing 1 Digit | https://proofwiki.org/wiki/Numbers_that_cannot_be_made_Prime_by_changing_1_Digit | https://proofwiki.org/wiki/Numbers_that_cannot_be_made_Prime_by_changing_1_Digit | [
"Prime Numbers",
"Recreational Mathematics",
"Numbers that cannot be made Prime by changing 1 Digit"
] | [
"Definition:Positive/Integer",
"Definition:Prime Number",
"Definition:Digit"
] | [
"Definition:Positive/Integer",
"Definition:Prime Number",
"Definition:Digit",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Positive/Integer",
"Definition:Composite Number",
"Definition:Prime Gap",
"Definition:Prime Number",
"Definition... |
proofwiki-13221 | Sum of Cubes of 3 Consecutive Integers which is Square | The following sequences of $3$ consecutive (strictly) positive integers have cubes that sum to a square:
:$1, 2, 3$
:$23, 24, 25$
No other such sequence of $3$ consecutive positive integers has the same property.
However, if we allow sequences containing zero and negative integers, we also have:
:$-1, 0, 1$
:$0, 1, 2$ | {{begin-eqn}}
{{eqn | l = 1^3 + 2^3 + 3^3
| r = 1 + 8 + 27
| c =
}}
{{eqn | r = 36
| c =
}}
{{eqn | r = 6^2
| c =
}}
{{eqn | l = 23^3 + 24^3 + 25^3
| r = 12 \, 167 + 13 \, 824 + 15 \, 625
| c =
}}
{{eqn | r = 41 \, 616
| c =
}}
{{eqn | r = 204^2
| c =
}}
{{eqn | l =... | The following [[Definition:Integer Sequence|sequences]] of $3$ consecutive [[Definition:Strictly Positive Integer|(strictly) positive integers]] have [[Definition:Cube Number|cubes]] that [[Definition:Integer Addition|sum]] to a [[Definition:Square Number|square]]:
:$1, 2, 3$
:$23, 24, 25$
No other such [[Definition... | {{begin-eqn}}
{{eqn | l = 1^3 + 2^3 + 3^3
| r = 1 + 8 + 27
| c =
}}
{{eqn | r = 36
| c =
}}
{{eqn | r = 6^2
| c =
}}
{{eqn | l = 23^3 + 24^3 + 25^3
| r = 12 \, 167 + 13 \, 824 + 15 \, 625
| c =
}}
{{eqn | r = 41 \, 616
| c =
}}
{{eqn | r = 204^2
| c =
}}
{{eqn | l =... | Sum of Cubes of 3 Consecutive Integers which is Square | https://proofwiki.org/wiki/Sum_of_Cubes_of_3_Consecutive_Integers_which_is_Square | https://proofwiki.org/wiki/Sum_of_Cubes_of_3_Consecutive_Integers_which_is_Square | [
"Sums of Cubes"
] | [
"Definition:Integer Sequence",
"Definition:Strictly Positive/Integer",
"Definition:Cube Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Integer Sequence",
"Definition:Zero (Number)",
"Definition:Negative... | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Cube Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Integer Sequence",
"Binomial Theorem/Examples/Cube of Sum",
"Binomial Theorem/Examples/Cube of Difference",
"Definition:Elliptic Curve",
"Definitio... |
proofwiki-13222 | Triangular Numbers which are also Pentagonal | The sequence of triangular numbers which are also pentagonal begins:
:$1, 210, 40 \, 755, 7 \, 906 \, 276, 1 \, 533 \, 776 \, 805, 297 \, 544 \, 793 \, 910, \ldots$
{{OEIS|A014979}} | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {3 \times 1 - 1} } 2
| c = Closed Form for Pentagonal Numbers
}}
{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
| c = Closed Form for Triangular Numbers
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 210
| r = \dfrac {12 \paren {3 \times 12 - 1} } 2
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Triangular Number|triangular numbers]] which are also [[Definition:Pentagonal Number|pentagonal]] begins:
:$1, 210, 40 \, 755, 7 \, 906 \, 276, 1 \, 533 \, 776 \, 805, 297 \, 544 \, 793 \, 910, \ldots$
{{OEIS|A014979}} | {{begin-eqn}}
{{eqn | l = 1
| r = \dfrac {1 \paren {3 \times 1 - 1} } 2
| c = [[Closed Form for Pentagonal Numbers]]
}}
{{eqn | r = \dfrac {1 \times \paren {1 + 1} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 210
| r = \dfrac {12 \paren {3 \times 12... | Triangular Numbers which are also Pentagonal | https://proofwiki.org/wiki/Triangular_Numbers_which_are_also_Pentagonal | https://proofwiki.org/wiki/Triangular_Numbers_which_are_also_Pentagonal | [
"Triangular Numbers",
"Pentagonal Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Triangular Number",
"Definition:Pentagonal Number"
] | [
"Closed Form for Pentagonal Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Pentagonal Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Pentagonal Numbers",
"Closed Form for Triangular Numbers"
] |
proofwiki-13223 | Numbers such that Divisor Count divides Phi divides Divisor Sum | The sequence of integers $n$ with the property that:
:$\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$
where:
:$\divides$ denotes divisibility
:$\sigma_0$ denotes the divisor count of $n$
:$\phi$ denotes the Euler $\phi$ (phi) function: the count of smaller integers coprime to $n$
:$\sigma_1$ denotes... | By inspection and investigation.
{{qed}} | The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] $n$ with the property that:
:$\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$
where:
:$\divides$ denotes [[Definition:Divisor of Integer|divisibility]]
:$\sigma_0$ denotes the [[Definition:Divisor Count Function|divisor c... | By inspection and investigation.
{{qed}} | Numbers such that Divisor Count divides Phi divides Divisor Sum | https://proofwiki.org/wiki/Numbers_such_that_Divisor_Count_divides_Phi_divides_Divisor_Sum | https://proofwiki.org/wiki/Numbers_such_that_Divisor_Count_divides_Phi_divides_Divisor_Sum | [
"Numbers such that Divisor Count divides Phi divides Divisor Sum",
"Divisor Count Function",
"Euler Phi Function",
"Divisor Sum Function"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Count Function",
"Definition:Euler Phi Function",
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Divisor Sum Function"
] | [] |
proofwiki-13224 | Number of Representations as Sum of Two Primes | The number of ways an integer $n$ can be represented as the sum of two primes is no greater than the number of primes in the interval $\closedint {\dfrac n 2} {n - 2}$. | Let $n = p + q$ where $p \le q$ and both $p$ and $q$ are primes.
There can be no more different representations of $n$ as $p + q$ than there are the number of possible options for $q$.
As $q \ge p$, it follows that $q \ge \dfrac n 2$.
Note that as $p \ge 2$, it follows that $q \le n - 2$.
The number of possible values ... | The number of ways an [[Definition:Integer|integer]] $n$ can be represented as the [[Definition:Integer Addition|sum]] of two [[Definition:Prime Number|primes]] is no greater than the number of [[Definition:Prime Number|primes]] in the [[Definition:Closed Real Interval|interval]] $\closedint {\dfrac n 2} {n - 2}$. | Let $n = p + q$ where $p \le q$ and both $p$ and $q$ are [[Definition:Prime Number|primes]].
There can be no more different representations of $n$ as $p + q$ than there are the number of possible options for $q$.
As $q \ge p$, it follows that $q \ge \dfrac n 2$.
Note that as $p \ge 2$, it follows that $q \le n - 2$.... | Number of Representations as Sum of Two Primes | https://proofwiki.org/wiki/Number_of_Representations_as_Sum_of_Two_Primes | https://proofwiki.org/wiki/Number_of_Representations_as_Sum_of_Two_Primes | [
"Prime Numbers"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Real Interval/Closed"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Real Interval/Closed"
] |
proofwiki-13225 | Integers whose Number of Representations as Sum of Two Primes is Maximum | $210$ is the largest integer which can be represented as the sum of two primes in the maximum number of ways.
The full list of such numbers is as follows:
:$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210$
{{OEIS|A141340}}
The list contains:
:$n \le 8$
:$n \le 18$ where $2 \divides n$
:$n \l... | From Number of Representations as Sum of Two Primes, the number of ways an integer $n$ can be represented as the sum of two primes is no greater than the number of primes in the interval $\closedint {\dfrac n 2} {n - 2}$.
The interval $\closedint {\dfrac {210} 2} {210 - 2}$ is $\closedint {105} {208}$.
The primes in th... | $210$ is the largest [[Definition:Integer|integer]] which can be represented as the [[Definition:Integer Addition|sum]] of two [[Definition:Prime Number|primes]] in the maximum number of ways.
The full list of such numbers is as follows:
:$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210$
{{... | From [[Number of Representations as Sum of Two Primes]], the number of ways an [[Definition:Integer|integer]] $n$ can be represented as the [[Definition:Integer Addition|sum]] of two [[Definition:Prime Number|primes]] is no greater than the number of [[Definition:Prime Number|primes]] in the [[Definition:Closed Real In... | Integers whose Number of Representations as Sum of Two Primes is Maximum | https://proofwiki.org/wiki/Integers_whose_Number_of_Representations_as_Sum_of_Two_Primes_is_Maximum | https://proofwiki.org/wiki/Integers_whose_Number_of_Representations_as_Sum_of_Two_Primes_is_Maximum | [
"Prime Numbers",
"Integers whose Number of Representations as Sum of Two Primes is Maximum"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Prime Number"
] | [
"Number of Representations as Sum of Two Primes",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Closed",
"Definition:Prime Number",
"Definition:Real Interval/Closed",
"Definit... |
proofwiki-13226 | 25 as Sum of 4 to 11 Squares | $25$ can be expressed as the sum of $n$ non-zero squares for all $n$ from $4$ to $11$. | We have:
{{begin-eqn}}
{{eqn | l = 25
| r = 4^2 + \paren {2 \times 2^2} + 1^2
| c = $4$ squares
}}
{{eqn | r = 3^2 + \paren {4 \times 2^2}
| c = $5$ squares
}}
{{eqn | r = \paren {2 \times 3^2} + 2^2 + \paren {3 \times 1^2}
| c = $6$ squares
}}
{{eqn | r = 4^2 + 2^2 + \paren {5 \times 1^2}
... | $25$ can be expressed as the [[Definition:Integer Addition|sum]] of $n$ non-zero [[Definition:Square Number|squares]] for all $n$ from $4$ to $11$. | We have:
{{begin-eqn}}
{{eqn | l = 25
| r = 4^2 + \paren {2 \times 2^2} + 1^2
| c = $4$ [[Definition:Square Number|squares]]
}}
{{eqn | r = 3^2 + \paren {4 \times 2^2}
| c = $5$ [[Definition:Square Number|squares]]
}}
{{eqn | r = \paren {2 \times 3^2} + 2^2 + \paren {3 \times 1^2}
| c = $6$ [[D... | 25 as Sum of 4 to 11 Squares | https://proofwiki.org/wiki/25_as_Sum_of_4_to_11_Squares | https://proofwiki.org/wiki/25_as_Sum_of_4_to_11_Squares | [
"Sums of Squares",
"25"
] | [
"Definition:Addition/Integers",
"Definition:Square Number"
] | [
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number"
] |
proofwiki-13227 | Ritz Method implies Not Worse Approximation with Increased Number of Functions | Consider the Ritz method.
Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$.
Let $J \sqbrk {\eta_n} = \mu_n$.
Then:
:$\mu_n \ge \mu_{n + 1}$ | Denote $\eta_{n + 1} = \eta_n + \alpha_{n + 1} \phi_{n + 1}$.
For $\alpha_{n + 1} = 0$ we have:
:$\eta_n = \eta_{n + 1}$
Suppose that $J \sqbrk {\eta_n}$ has been minimised {{WRT}} $\boldsymbol \alpha$.
If:
:$\exists \boldsymbol \alpha \in \R^n: \nexists \alpha_{n + 1} \ne 0: J \sqbrk {\eta_n} > J \sqbrk {\eta_{n + 1} ... | Consider the [[Definition:Ritz Method|Ritz method]].
Let $\eta_n = \boldsymbol \alpha \boldsymbol \phi$.
Let $J \sqbrk {\eta_n} = \mu_n$.
Then:
:$\mu_n \ge \mu_{n + 1}$ | Denote $\eta_{n + 1} = \eta_n + \alpha_{n + 1} \phi_{n + 1}$.
For $\alpha_{n + 1} = 0$ we have:
:$\eta_n = \eta_{n + 1}$
Suppose that $J \sqbrk {\eta_n}$ has been [[Definition:Minimum Value|minimised]] {{WRT}} $\boldsymbol \alpha$.
If:
:$\exists \boldsymbol \alpha \in \R^n: \nexists \alpha_{n + 1} \ne 0: J \sqbrk {... | Ritz Method implies Not Worse Approximation with Increased Number of Functions | https://proofwiki.org/wiki/Ritz_Method_implies_Not_Worse_Approximation_with_Increased_Number_of_Functions | https://proofwiki.org/wiki/Ritz_Method_implies_Not_Worse_Approximation_with_Increased_Number_of_Functions | [
"Calculus of Variations"
] | [
"Definition:Ritz Method"
] | [
"Definition:Minimum Value of Real Function/Absolute",
"Definition:Minimum Value of Real Function/Absolute"
] |
proofwiki-13228 | Dissection of Cube into 3 Cubes using 8 Pieces | A cube can be dissected into $3$ smaller cubes by cutting it into $8$ pieces and reassembling them. | {{ProofWanted|Apparently it can be done using the result Cubes which are Sum of Three Cubes.}} | A [[Definition:Cube (Geometry)|cube]] can be [[Definition:Dissection|dissected]] into $3$ smaller [[Definition:Cube (Geometry)|cubes]] by cutting it into $8$ pieces and reassembling them. | {{ProofWanted|Apparently it can be done using the result [[Cubes which are Sum of Three Cubes]].}} | Dissection of Cube into 3 Cubes using 8 Pieces | https://proofwiki.org/wiki/Dissection_of_Cube_into_3_Cubes_using_8_Pieces | https://proofwiki.org/wiki/Dissection_of_Cube_into_3_Cubes_using_8_Pieces | [
"Dissections"
] | [
"Definition:Cube/Geometry",
"Definition:Dissection",
"Definition:Cube/Geometry"
] | [
"Cubes which are Sum of Three Cubes"
] |
proofwiki-13229 | Smallest Magic Constant of Order 3 Multiplicative Magic Square | The magic constant of the smallest possible multiplicative magic square with the smallest magic constant is as follows.
{{:Multiplicative Magic Square/Examples/Order 3/Smallest}}
</onlyinclude> | From Smallest Multiplicative Magic Square is of Order 3, the smallest possible multiplicative magic square is of order $3$.
Let $C$ be a magic constant for an order $3$ multiplicative magic square.
We prove that $C \ge 216$ is smallest in $2$ steps:
:Step $1$: If $C$ is a prime power $p^k$, then $k \ge 8$.
:Step $2$: F... | The [[Definition:Multiplicative Magic Constant|magic constant]] of the smallest possible [[Definition:Multiplicative Magic Square|multiplicative magic square]] with the smallest [[Definition:Multiplicative Magic Constant|magic constant]] is as follows.
{{:Multiplicative Magic Square/Examples/Order 3/Smallest}}
</onlyin... | From [[Smallest Multiplicative Magic Square is of Order 3]], the smallest possible [[Definition:Multiplicative Magic Square|multiplicative magic square]] is of [[Definition:Order of Multiplicative Magic Square|order $3$]].
Let $C$ be a [[Definition:Multiplicative Magic Constant|magic constant]] for an [[Definition:Or... | Smallest Magic Constant of Order 3 Multiplicative Magic Square | https://proofwiki.org/wiki/Smallest_Magic_Constant_of_Order_3_Multiplicative_Magic_Square | https://proofwiki.org/wiki/Smallest_Magic_Constant_of_Order_3_Multiplicative_Magic_Square | [
"Multiplicative Magic Squares"
] | [
"Definition:Multiplicative Magic Square/Magic Constant",
"Definition:Multiplicative Magic Square",
"Definition:Multiplicative Magic Square/Magic Constant"
] | [
"Smallest Multiplicative Magic Square is of Order 3",
"Definition:Multiplicative Magic Square",
"Definition:Multiplicative Magic Square/Order",
"Definition:Multiplicative Magic Square/Magic Constant",
"Definition:Multiplicative Magic Square/Order",
"Definition:Multiplicative Magic Square",
"Definition:P... |
proofwiki-13230 | Smallest Multiplicative Magic Square is of Order 3 | The order of the smallest multiplicative magic square is $3$, for example:
{{:Multiplicative Magic Square/Examples/Order 3/Smallest}} | Suppose there were an order $2$ multiplicative magic square $M$.
Let $a$ be the element of row $1$ and column $1$.
Let $a b$ be the magic constant of $M$.
Then $b$ is:
:the element of row $1$ and column $2$, to make the product of row $1$ equal to $a b$
:the element of row $2$ and column $1$, to make the product of col... | The [[Definition:Order of Multiplicative Magic Square|order]] of the smallest [[Definition:Multiplicative Magic Square|multiplicative magic square]] is $3$, for example:
{{:Multiplicative Magic Square/Examples/Order 3/Smallest}} | Suppose there were an [[Definition:Order of Multiplicative Magic Square|order $2$]] [[Definition:Multiplicative Magic Square|multiplicative magic square]] $M$.
Let $a$ be the [[Definition:Element of Array|element]] of [[Definition:Row of Array|row $1$]] and [[Definition:Column of Array|column $1$]].
Let $a b$ be the ... | Smallest Multiplicative Magic Square is of Order 3 | https://proofwiki.org/wiki/Smallest_Multiplicative_Magic_Square_is_of_Order_3 | https://proofwiki.org/wiki/Smallest_Multiplicative_Magic_Square_is_of_Order_3 | [
"Multiplicative Magic Squares"
] | [
"Definition:Multiplicative Magic Square/Order",
"Definition:Multiplicative Magic Square"
] | [
"Definition:Multiplicative Magic Square/Order",
"Definition:Multiplicative Magic Square",
"Definition:Array/Element",
"Definition:Array/Row",
"Definition:Array/Column",
"Definition:Multiplicative Magic Square/Magic Constant",
"Definition:Array/Element",
"Definition:Array/Row",
"Definition:Array/Colu... |
proofwiki-13231 | For Complete Ritz Sequence Continuous Functional approaches its Minimal Value | Let $J$ be a continuous functional.
Let $\sequence {\phi_n}$ be a complete Ritz sequence.
{{explain|The concept of $\sequence {\phi_n}$ does not appear to be related in any way to the statement of the theorem.}}
Then:
:$\ds \lim_{n \mathop \to \infty} \mu_n = \mu$
where $\ds \mu = \inf_y J \sqbrk y$. | Let $y^*: \R \to \R$ be such that:
:$\forall \epsilon > 0: J \sqbrk {y^*} < \mu + \epsilon$
By assumption of continuity of $J$:
:$\forall \epsilon > 0: \exists \map \delta \epsilon > 0: \paren {\size {y - y^*} < \delta} \implies \paren {\size {J \sqbrk y - J \sqbrk {y^*} } < \epsilon}$
Let $\eta_n = \bsalpha \bsphi$, s... | Let $J$ be a [[Definition:Continuous Functional|continuous functional]].
Let $\sequence {\phi_n}$ be a [[Definition:Complete Ritz Sequence|complete Ritz sequence]].
{{explain|The concept of $\sequence {\phi_n}$ does not appear to be related in any way to the statement of the theorem.}}
Then:
:$\ds \lim_{n \mathop \... | Let $y^*: \R \to \R$ be such that:
:$\forall \epsilon > 0: J \sqbrk {y^*} < \mu + \epsilon$
By assumption of [[Definition:Continuous Functional|continuity of $J$]]:
:$\forall \epsilon > 0: \exists \map \delta \epsilon > 0: \paren {\size {y - y^*} < \delta} \implies \paren {\size {J \sqbrk y - J \sqbrk {y^*} } < \eps... | For Complete Ritz Sequence Continuous Functional approaches its Minimal Value | https://proofwiki.org/wiki/For_Complete_Ritz_Sequence_Continuous_Functional_approaches_its_Minimal_Value | https://proofwiki.org/wiki/For_Complete_Ritz_Sequence_Continuous_Functional_approaches_its_Minimal_Value | [
"Calculus of Variations"
] | [
"Definition:Continuity/Functional",
"Definition:Complete Ritz Sequence"
] | [
"Definition:Continuity/Functional",
"Definition:Dimension of Vector Space/Vector",
"Definition:Real Vector Space",
"Definition:Inequality",
"Definition:Finite",
"Definition:Continuity/Functional",
"Definition:Inequality"
] |
proofwiki-13232 | 219 Fedorov Groups | There are $219$ Fedorov groups, if chiral copies are considered the same. | {{ProofWanted|First the concept of Fedorov group needs to be explored.}} | There are $219$ [[Definition:Fedorov Group|Fedorov groups]], if [[Definition:Chirality|chiral copies]] are considered the same. | {{ProofWanted|First the concept of [[Definition:Fedorov Group|Fedorov group]] needs to be explored.}} | 219 Fedorov Groups | https://proofwiki.org/wiki/219_Fedorov_Groups | https://proofwiki.org/wiki/219_Fedorov_Groups | [
"Fedorov Groups",
"219"
] | [
"Definition:Fedorov Group",
"Definition:Chirality"
] | [
"Definition:Fedorov Group"
] |
proofwiki-13233 | 17 Wallpaper Groups | There are $17$ wallpaper groups. | {{ProofWanted|First the concept of wallpaper group needs to be explored.}} | There are $17$ [[Definition:Wallpaper Group|wallpaper groups]]. | {{ProofWanted|First the concept of [[Definition:Wallpaper Group|wallpaper group]] needs to be explored.}} | 17 Wallpaper Groups | https://proofwiki.org/wiki/17_Wallpaper_Groups | https://proofwiki.org/wiki/17_Wallpaper_Groups | [
"Wallpaper Groups",
"17"
] | [
"Definition:Wallpaper Group"
] | [
"Definition:Wallpaper Group"
] |
proofwiki-13234 | 230 Fedorov Groups where Chiral Pairs are Distinct | There are $230$ Fedorov groups, if chiral copies are considered distinct. | {{ProofWanted|First the concept of Fedorov group needs to be explored.}} | There are $230$ [[Definition:Fedorov Group|Fedorov groups]], if [[Definition:Chirality|chiral copies]] are considered [[Definition:Distinct|distinct]]. | {{ProofWanted|First the concept of [[Definition:Fedorov Group|Fedorov group]] needs to be explored.}} | 230 Fedorov Groups where Chiral Pairs are Distinct | https://proofwiki.org/wiki/230_Fedorov_Groups_where_Chiral_Pairs_are_Distinct | https://proofwiki.org/wiki/230_Fedorov_Groups_where_Chiral_Pairs_are_Distinct | [
"Fedorov Groups",
"230"
] | [
"Definition:Fedorov Group",
"Definition:Chirality",
"Definition:Distinct"
] | [
"Definition:Fedorov Group"
] |
proofwiki-13235 | Equivalence of Definitions of Amicable Pair | Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.
{{TFAE|def = Amicable Pair}} | === Definition 1 is equivalent to Definition 2 ===
Let $\map s n$ denote the aliquot sum of (strictly) positive integer $n$.
The sum of all the divisors of a (strictly) positive integer $n$ is $\map {\sigma_1} n$, where $\sigma_1$ is the divisor sum function.
The aliquot sum of $n$ is the sum of the divisors of $n$ wit... | Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
{{TFAE|def = Amicable Pair}} | === Definition 1 is equivalent to Definition 2 ===
Let $\map s n$ denote the [[Definition:Aliquot Sum|aliquot sum]] of [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$.
The [[Definition:Integer Addition|sum]] of all the [[Definition:Divisor of Integer|divisors]] of a [[Definition:Strictly Posi... | Equivalence of Definitions of Amicable Pair | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Amicable_Pair | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Amicable_Pair | [
"Amicable Pairs"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Definition:Aliquot Sum",
"Definition:Strictly Positive/Integer",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Aliquot Sum",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)... |
proofwiki-13236 | Thabit's Rule | Let $n$ be a positive integer such that:
{{begin-eqn}}
{{eqn | l = a
| r = 3 \times 2^n - 1
| c =
}}
{{eqn | l = b
| r = 3 \times 2^{n - 1} - 1
| c =
}}
{{eqn | l = c
| r = 9 \times 2^{2 n - 1} - 1
| c =
}}
{{end-eqn}}
are all prime.
Then:
:$\tuple {2^n a b, 2^n c}$
forms an amica... | Let $r = 2^n a b, s = 2^n c$.
Let $\map {\sigma_1} k$ denote the divisor sum of an integer $k$.
From Divisor Sum of Power of 2:
:$\map {\sigma_1} {2^n} = 2^{n + 1} - 1$
From Divisor Sum of Prime Number:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} a
| r = 3 \times 2^n
| c =
}}
{{eqn | l = \map {\sigma_1} b
... | Let $n$ be a [[Definition:Positive Integer|positive integer]] such that:
{{begin-eqn}}
{{eqn | l = a
| r = 3 \times 2^n - 1
| c =
}}
{{eqn | l = b
| r = 3 \times 2^{n - 1} - 1
| c =
}}
{{eqn | l = c
| r = 9 \times 2^{2 n - 1} - 1
| c =
}}
{{end-eqn}}
are all [[Definition:Prime N... | Let $r = 2^n a b, s = 2^n c$.
Let $\map {\sigma_1} k$ denote the [[Definition:Divisor Sum Function|divisor sum]] of an [[Definition:Integer|integer]] $k$.
From [[Divisor Sum of Power of 2]]:
:$\map {\sigma_1} {2^n} = 2^{n + 1} - 1$
From [[Divisor Sum of Prime Number]]:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} a
... | Thabit's Rule | https://proofwiki.org/wiki/Thabit's_Rule | https://proofwiki.org/wiki/Thabit's_Rule | [
"Amicable Pairs"
] | [
"Definition:Positive/Integer",
"Definition:Prime Number",
"Definition:Amicable Pair"
] | [
"Definition:Divisor Sum Function",
"Definition:Integer",
"Divisor Sum of Power of 2",
"Divisor Sum of Prime Number",
"Divisor Sum Function is Multiplicative",
"Definition:Amicable Pair"
] |
proofwiki-13237 | Smaller of Thabit Pair is Tetrahedral | Let $\tuple {m_1, m_2}$ be a Thabit pair such that $m_1 < m_2$.
Then $m_1$ is a tetrahedral number. | By Thabit's Rule:
{{begin-eqn}}
{{eqn | l = m_1
| r = 2^n \times \paren {3 \times 2^{n - 1} - 1} \times \paren {3 \times 2^n - 1}
| c =
}}
{{eqn | l = m_2
| r = 2^n \times \paren {9 \times 2^{2 n - 1} - 1}
| c =
}}
{{end-eqn}}
for some $n \in \Z_{\ge 0}$.
We have that:
{{begin-eqn}}
{{eqn | l ... | Let $\tuple {m_1, m_2}$ be a [[Definition:Thabit Pair|Thabit pair]] such that $m_1 < m_2$.
Then $m_1$ is a [[Definition:Tetrahedral Number|tetrahedral number]]. | By [[Thabit's Rule]]:
{{begin-eqn}}
{{eqn | l = m_1
| r = 2^n \times \paren {3 \times 2^{n - 1} - 1} \times \paren {3 \times 2^n - 1}
| c =
}}
{{eqn | l = m_2
| r = 2^n \times \paren {9 \times 2^{2 n - 1} - 1}
| c =
}}
{{end-eqn}}
for some $n \in \Z_{\ge 0}$.
We have that:
{{begin-eqn}}
{{... | Smaller of Thabit Pair is Tetrahedral | https://proofwiki.org/wiki/Smaller_of_Thabit_Pair_is_Tetrahedral | https://proofwiki.org/wiki/Smaller_of_Thabit_Pair_is_Tetrahedral | [
"Amicable Pairs",
"Tetrahedral Numbers"
] | [
"Definition:Thabit Pair",
"Definition:Tetrahedral Number"
] | [
"Thabit's Rule",
"Closed Form for Tetrahedral Numbers",
"Definition:Tetrahedral Number"
] |
proofwiki-13238 | Sturm-Liouville Problem | Let:
:$P \in C^\infty \map P x > 0$
:$Q \in C^0$
:$-\paren {P y'}' + Q y = \lambda y$
:$\map y a = \map y b = 0$
Then the Sturm-Liouville problem has an infinite sequence of eigenvalues $\set {\lambda^{\paren n} }$, and to each $\lambda^{\paren n}$ corresponds an eigenfunction $y^{\paren n}$, unique up to a constant fa... | :$J \sqbrk y = \ds \int_a^b \paren {P y'^2 + Q y^2} \rd x$
:$\ds \int_a^b y^2 \rd x = 1$
:$\ds \int_a^b \paren {P y'^2 + Q y^2} \rd x > \int_a^b Q y^2 \rd x \ge M \int_a^b y^2 \rd x = M$
:$M = \min \limits_{a \mathop \le x \mathop \le b} \map Q x$
Assume $a = 0$, $b = \pi$.
Choose $\set {\map {\phi_n} x} = \set {\sin n... | Let:
:$P \in C^\infty \map P x > 0$
:$Q \in C^0$
:$-\paren {P y'}' + Q y = \lambda y$
:$\map y a = \map y b = 0$
Then the Sturm-Liouville problem has an infinite sequence of eigenvalues $\set {\lambda^{\paren n} }$, and to each $\lambda^{\paren n}$ corresponds an eigenfunction $y^{\paren n}$, unique up to a consta... | :$J \sqbrk y = \ds \int_a^b \paren {P y'^2 + Q y^2} \rd x$
:$\ds \int_a^b y^2 \rd x = 1$
:$\ds \int_a^b \paren {P y'^2 + Q y^2} \rd x > \int_a^b Q y^2 \rd x \ge M \int_a^b y^2 \rd x = M$
:$M = \min \limits_{a \mathop \le x \mathop \le b} \map Q x$
Assume $a = 0$, $b = \pi$.
Choose $\set {\map {\phi_n} x} = \set {\... | Sturm-Liouville Problem | https://proofwiki.org/wiki/Sturm-Liouville_Problem | https://proofwiki.org/wiki/Sturm-Liouville_Problem | [] | [] | [] |
proofwiki-13239 | Numbers in Even-Even Amicable Pair are not Divisible by 3 | Let $\tuple {m_1, m_2}$ be an amicable pair such that both $m_1$ and $m_2$ are even.
Then neither $m_1$ nor $m_2$ is divisible by $3$. | An amicable pair must be formed from a smaller abundant number and a larger deficient number.
Suppose both $m_1, m_2$ are divisible by $3$.
Since both are even, they must also be divisible by $6$.
However $6$ is a perfect number.
By Multiple of Perfect Number is Abundant, neither can be deficient.
So $m_1, m_2$ cannot ... | Let $\tuple {m_1, m_2}$ be an [[Definition:Amicable Pair|amicable pair]] such that both $m_1$ and $m_2$ are [[Definition:Even Integer|even]].
Then neither $m_1$ nor $m_2$ is [[Definition:Divisor of Integer|divisible]] by $3$. | An [[Definition:Amicable Pair|amicable pair]] must be formed from a smaller [[Definition:Abundant Number|abundant number]] and a larger [[Definition:Deficient Number|deficient number]].
Suppose both $m_1, m_2$ are [[Definition:Divisor of Integer|divisible]] by $3$.
Since both are [[Definition:Even Integer|even]], the... | Numbers in Even-Even Amicable Pair are not Divisible by 3 | https://proofwiki.org/wiki/Numbers_in_Even-Even_Amicable_Pair_are_not_Divisible_by_3 | https://proofwiki.org/wiki/Numbers_in_Even-Even_Amicable_Pair_are_not_Divisible_by_3 | [
"Amicable Pairs"
] | [
"Definition:Amicable Pair",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Amicable Pair",
"Definition:Abundant Number",
"Definition:Deficient Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Perfect Number",
"Multiple of Perfect Number is Abundant",
"Definition:Deficient Number",
... |
proofwiki-13240 | Amicable Pair with Smallest Common Prime Factor 5 | An amicable pair whose smallest common prime factor is greater than $3$ has the elements:
:$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8693 \times 19 \, 479 \, 382 \, 229$
and:
:$m_2 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23... | It is to be demonstrated that these numbers are amicable.
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
is the prime decomposition... | An [[Definition:Amicable Pair|amicable pair]] whose smallest [[Definition:Common Divisor of Integers|common]] [[Definition:Prime Factor|prime factor]] is greater than $3$ has the elements:
:$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8693 \times 19 \,... | It is to be demonstrated that these numbers are [[Definition:Amicable Pair|amicable]].
From [[Divisor Sum of Integer]]:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r... | Amicable Pair with Smallest Common Prime Factor 5 | https://proofwiki.org/wiki/Amicable_Pair_with_Smallest_Common_Prime_Factor_5 | https://proofwiki.org/wiki/Amicable_Pair_with_Smallest_Common_Prime_Factor_5 | [
"Amicable Pairs"
] | [
"Definition:Amicable Pair",
"Definition:Common Divisor/Integers",
"Definition:Prime Factor",
"Definition:Amicable Pair",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Amicable Pair",
"Divisor Sum of Integer",
"Definition:Prime Decomposition",
"Definition:Divisor Sum Function",
"Definition:Prime Factor",
"Definition:Prime Factor",
"Divisor Sum of Prime Number",
"Divisor Sum of Power of Prime",
"Divisor Sum of Power of Prime",
"Divisor Sum of Prime Nu... |
proofwiki-13241 | Equivalence of Definitions of Amicable Triplet | Let $m_1, m_2, m_3 \in \Z_{>0}$ be (strictly) positive integers.
{{TFAE|def = Amicable Triplet}} | For $n \in \Z_{>0}$, let $\map s n$ denote the aliquot sum of (strictly) positive integer $n$.
The sum of all the divisors of a (strictly) positive integer $n$ is $\map {\sigma_1} n$, where $\sigma_1$ is the divisor sum function.
The aliquot sum of $n$ is the sum of the divisors of $n$ with $n$ excluded.
Thus:
:$\map s... | Let $m_1, m_2, m_3 \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
{{TFAE|def = Amicable Triplet}} | For $n \in \Z_{>0}$, let $\map s n$ denote the [[Definition:Aliquot Sum|aliquot sum]] of [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$.
The [[Definition:Integer Addition|sum]] of all the [[Definition:Divisor of Integer|divisors]] of a [[Definition:Strictly Positive Integer|(strictly) positiv... | Equivalence of Definitions of Amicable Triplet | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Amicable_Triplet | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Amicable_Triplet | [
"Amicable Triplets"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Definition:Aliquot Sum",
"Definition:Strictly Positive/Integer",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Divisor Sum Function",
"Definition:Aliquot Sum",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)... |
proofwiki-13242 | Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers | The largest positive integer which cannot be expressed as the sum of less than $37$ positive fifth powers is $223$:
:$223 = 31 \times 1^5 + 6 \times 2^5$ | {{ProofWanted|It needs to be shown that there are no larger numbers with this property.}} | The largest [[Definition:Positive Integer|positive integer]] which cannot be expressed as the [[Definition:Integer Addition|sum]] of less than $37$ [[Definition:Positive Integer|positive]] [[Definition:Fifth Power|fifth powers]] is $223$:
:$223 = 31 \times 1^5 + 6 \times 2^5$ | {{ProofWanted|It needs to be shown that there are no larger numbers with this property.}} | Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Less_than_37_Positive_Fifth_Powers | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Less_than_37_Positive_Fifth_Powers | [
"Fifth Powers",
"Hilbert-Waring Theorem",
"223"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Positive/Integer",
"Definition:Fifth Power"
] | [] |
proofwiki-13243 | Smallest Triple of Consecutive Sums of Squares | The smallest triple of consecutive positive integers each of which is the sum of two squares is:
:$\tuple {232, 233, 234}$ | We have:
{{begin-eqn}}
{{eqn | l = 232
| r = 14^2 + 6^2
| c =
}}
{{eqn | l = 233
| r = 13^2 + 8^2
| c =
}}
{{eqn | l = 234
| r = 15^2 + 3^2
| c =
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown this is the smallest such triple.}} | The smallest [[Definition:Ordered Triple|triple]] of consecutive [[Definition:Positive Integer|positive integers]] each of which is the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]] is:
:$\tuple {232, 233, 234}$ | We have:
{{begin-eqn}}
{{eqn | l = 232
| r = 14^2 + 6^2
| c =
}}
{{eqn | l = 233
| r = 13^2 + 8^2
| c =
}}
{{eqn | l = 234
| r = 15^2 + 3^2
| c =
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown this is the smallest such triple.}} | Smallest Triple of Consecutive Sums of Squares | https://proofwiki.org/wiki/Smallest_Triple_of_Consecutive_Sums_of_Squares | https://proofwiki.org/wiki/Smallest_Triple_of_Consecutive_Sums_of_Squares | [
"Sums of Squares"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number"
] | [] |
proofwiki-13244 | No Quadruple of Consecutive Sums of Squares Exists | It is not possible for a quadruple of consecutive positive integers each of which is the sum of two squares. | $4$ consecutive positive integers will be in the forms:
{{begin-eqn}}
{{eqn | l = n_0
| o = \equiv
| r = 0
| rr= \pmod 4
| c =
}}
{{eqn | l = n_1
| o = \equiv
| r = 1
| rr= \pmod 4
| c =
}}
{{eqn | l = n_2
| o = \equiv
| r = 2
| rr= \pmod 4
| c =... | It is not possible for a [[Definition:Ordered Quadruple|quadruple]] of consecutive [[Definition:Positive Integer|positive integers]] each of which is the [[Definition:Integer Addition|sum]] of two [[Definition:Square Number|squares]]. | $4$ consecutive [[Definition:Positive Integer|positive integers]] will be in the forms:
{{begin-eqn}}
{{eqn | l = n_0
| o = \equiv
| r = 0
| rr= \pmod 4
| c =
}}
{{eqn | l = n_1
| o = \equiv
| r = 1
| rr= \pmod 4
| c =
}}
{{eqn | l = n_2
| o = \equiv
| r = ... | No Quadruple of Consecutive Sums of Squares Exists | https://proofwiki.org/wiki/No_Quadruple_of_Consecutive_Sums_of_Squares_Exists | https://proofwiki.org/wiki/No_Quadruple_of_Consecutive_Sums_of_Squares_Exists | [
"Sums of Squares"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Quadruple",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number"
] | [
"Definition:Positive/Integer",
"Sum of Two Squares not Congruent to 3 modulo 4",
"Definition:Addition/Integers",
"Definition:Square Number"
] |
proofwiki-13245 | Directed iff Filtered in Dual Ordered Set | Let $\struct {S, \preceq_1}$ be an ordered set.
Let $\struct {S, \preceq_2}$ be a dual ordered set of $\struct {S, \preceq_1}$
Let $X \subseteq S$.
Then:
:$X$ is directed in $\struct {S, \preceq_1}$
{{iff}}:
:$X$ is filtered in $\struct {S, \preceq_2}$. | === Sufficient Condition ===
Assume that:
:$X$ is directed in $\struct {S, \preceq_1}$
Thus $X$ is non-empty.
Let $x, y \in S$.
By definition of directed:
:$\exists z \in S: x \preceq_1 z \land y \preceq_1 z$
Thus by definition of dual ordered set:
:$z \preceq_2 x \land z \preceq_2 y$
{{qed|lemma}} | Let $\struct {S, \preceq_1}$ be an [[Definition:Ordered Set|ordered set]].
Let $\struct {S, \preceq_2}$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $\struct {S, \preceq_1}$
Let $X \subseteq S$.
Then:
:$X$ is [[Definition:Directed Subset|directed]] in $\struct {S, \preceq_1}$
{{iff}}:
:$X$ is [[Definiti... | === Sufficient Condition ===
Assume that:
:$X$ is [[Definition:Directed Subset|directed]] in $\struct {S, \preceq_1}$
Thus $X$ is [[Definition:Non-Empty Set|non-empty]].
Let $x, y \in S$.
By definition of [[Definition:Directed Subset|directed]]:
:$\exists z \in S: x \preceq_1 z \land y \preceq_1 z$
Thus by definit... | Directed iff Filtered in Dual Ordered Set | https://proofwiki.org/wiki/Directed_iff_Filtered_in_Dual_Ordered_Set | https://proofwiki.org/wiki/Directed_iff_Filtered_in_Dual_Ordered_Set | [
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Directed Subset",
"Definition:Filtered Subset"
] | [
"Definition:Directed Subset",
"Definition:Non-Empty Set",
"Definition:Directed Subset",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Non-Empty Set",
"Definition:Dual Ordering/Dual Ordered Set"
] |
proofwiki-13246 | Sum of Two Squares not Congruent to 3 modulo 4 | Let $n \in \Z$ such that $n = a^2 + b^2$ where $a, b \in \Z$.
Then $n$ is not congruent modulo $4$ to $3$. | Let $n \equiv 3 \pmod 4$.
{{AimForCont}} $n$ can be expressed as the sum of two squares:
:$n = a^2 + b^2$.
From Square Modulo 4, either $a^2 \equiv 0$ or $a^2 \equiv 1 \pmod 4$.
Similarly for $b^2$.
So $a^2 + b^2 \not \equiv 3 \pmod 4$ whatever $a$ and $b$ are.
Thus $n$ cannot be the sum of two squares.
The result foll... | Let $n \in \Z$ such that $n = a^2 + b^2$ where $a, b \in \Z$.
Then $n$ is not [[Definition:Congruence Modulo Integer|congruent modulo $4$]] to $3$. | Let $n \equiv 3 \pmod 4$.
{{AimForCont}} $n$ can be expressed as the sum of two [[Definition:Square Number|squares]]:
:$n = a^2 + b^2$.
From [[Square Modulo 4]], either $a^2 \equiv 0$ or $a^2 \equiv 1 \pmod 4$.
Similarly for $b^2$.
So $a^2 + b^2 \not \equiv 3 \pmod 4$ whatever $a$ and $b$ are.
Thus $n$ cannot be t... | Sum of Two Squares not Congruent to 3 modulo 4 | https://proofwiki.org/wiki/Sum_of_Two_Squares_not_Congruent_to_3_modulo_4 | https://proofwiki.org/wiki/Sum_of_Two_Squares_not_Congruent_to_3_modulo_4 | [
"Sums of Squares"
] | [
"Definition:Congruence (Number Theory)/Integers"
] | [
"Definition:Square Number",
"Square Modulo 4",
"Definition:Addition/Integers",
"Definition:Square Number",
"Proof by Contradiction",
"Category:Sums of Squares"
] |
proofwiki-13247 | Numbers not Expressible as Sum of Less than 9 Positive Cubes | The following are the only positive integers cannot be expressed as the sum of less than $9$ positive cubes:
{{begin-eqn}}
{{eqn | l = 23
| r = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
| c =
}}
{{eqn | l = 239
| r = 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3
| c =
}}
{{end-... | {{ProofWanted|It needs to be demonstrated that these are the only ones.}} | The following are the only [[Definition:Positive Integer|positive integers]] cannot be expressed as the [[Definition:Integer Addition|sum]] of less than $9$ [[Definition:Positive Integer|positive]] [[Definition:Cube Number|cubes]]:
{{begin-eqn}}
{{eqn | l = 23
| r = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 ... | {{ProofWanted|It needs to be demonstrated that these are the only ones.}} | Numbers not Expressible as Sum of Less than 9 Positive Cubes | https://proofwiki.org/wiki/Numbers_not_Expressible_as_Sum_of_Less_than_9_Positive_Cubes | https://proofwiki.org/wiki/Numbers_not_Expressible_as_Sum_of_Less_than_9_Positive_Cubes | [
"Hilbert-Waring Theorem",
"Cube Numbers",
"23",
"239"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Positive/Integer",
"Definition:Cube Number"
] | [] |
proofwiki-13248 | Largest Prime Factor of n squared plus 1 | Let $n \in \Z$ be greater than $239$.
Then the largest prime factor of $n^2 + 1$ is at least $17$. | We note that for $n = 239$ we have:
{{begin-eqn}}
{{eqn | r = 239^2 + 1
| o =
| c =
}}
{{eqn | r = 57122
| c =
}}
{{eqn | r = 2 \times 13^4
| c =
}}
{{end-eqn}}
Thus the largest prime factor of $239^2 + 1$ is $13$.
{{ProofWanted|Now the rest of the result is to be shown.}} | Let $n \in \Z$ be greater than $239$.
Then the largest [[Definition:Prime Factor|prime factor]] of $n^2 + 1$ is at least $17$. | We note that for $n = 239$ we have:
{{begin-eqn}}
{{eqn | r = 239^2 + 1
| o =
| c =
}}
{{eqn | r = 57122
| c =
}}
{{eqn | r = 2 \times 13^4
| c =
}}
{{end-eqn}}
Thus the largest [[Definition:Prime Factor|prime factor]] of $239^2 + 1$ is $13$.
{{ProofWanted|Now the rest of the result is to... | Largest Prime Factor of n squared plus 1 | https://proofwiki.org/wiki/Largest_Prime_Factor_of_n_squared_plus_1 | https://proofwiki.org/wiki/Largest_Prime_Factor_of_n_squared_plus_1 | [
"Number Theory",
"239"
] | [
"Definition:Prime Factor"
] | [
"Definition:Prime Factor"
] |
proofwiki-13249 | Solution of Ljunggren Equation | The only solutions of the Ljunggren equation:
:$x^2 + 1 = 2 y^4$
are:
:$x = 1, y = 1$
:$x = 239, y = 13$
{{OEIS|A229384}} | Setting $x = 1$:
{{begin-eqn}}
{{eqn | r = 1^2 + 1
| o =
| c =
}}
{{eqn | r = 2
| c =
}}
{{eqn | r = 2 \times 1^4
| c =
}}
{{end-eqn}}
and so $y = 1$.
Setting $x = 239$:
{{begin-eqn}}
{{eqn | r = 239^2 + 1
| o =
| c =
}}
{{eqn | r = 57122
| c =
}}
{{eqn | r = 2 \times 13... | The only solutions of the [[Definition:Ljunggren Equation|Ljunggren equation]]:
:$x^2 + 1 = 2 y^4$
are:
:$x = 1, y = 1$
:$x = 239, y = 13$
{{OEIS|A229384}} | Setting $x = 1$:
{{begin-eqn}}
{{eqn | r = 1^2 + 1
| o =
| c =
}}
{{eqn | r = 2
| c =
}}
{{eqn | r = 2 \times 1^4
| c =
}}
{{end-eqn}}
and so $y = 1$.
Setting $x = 239$:
{{begin-eqn}}
{{eqn | r = 239^2 + 1
| o =
| c =
}}
{{eqn | r = 57122
| c =
}}
{{eqn | r = 2 \tim... | Solution of Ljunggren Equation | https://proofwiki.org/wiki/Solution_of_Ljunggren_Equation | https://proofwiki.org/wiki/Solution_of_Ljunggren_Equation | [
"Diophantine Equations",
"239",
"13"
] | [
"Definition:Ljunggren Equation"
] | [
"Largest Prime Factor of n squared plus 1"
] |
proofwiki-13250 | Solutions of Diophantine Equation x^4 + y^4 = z^2 + 1 for x = 239 | Consider the indeterminate Diophantine equation:
:$x^4 + y^4 = z^2 + 1$
When $x = 239$ and $x > y$, there are $3$ solutions:
{{begin-eqn}}
{{eqn | l = 239^4 + 104^4
| r = 58 \, 136^2 + 1
| c =
}}
{{eqn | l = 239^4 + 143^4
| r = 60 \, 671^2 + 1
| c =
}}
{{eqn | l = 239^4 + 208^4
| r = 71 ... | {{ProofWanted|It remains to be shown there are no other solutions. This can be achieved by brute force. For this page, it is then adequate to present a pseudocode program to illustrate it.}} | Consider the [[Definition:Indeterminate Equation|indeterminate]] [[Definition:Diophantine Equation|Diophantine equation]]:
:$x^4 + y^4 = z^2 + 1$
When $x = 239$ and $x > y$, there are $3$ solutions:
{{begin-eqn}}
{{eqn | l = 239^4 + 104^4
| r = 58 \, 136^2 + 1
| c =
}}
{{eqn | l = 239^4 + 143^4
| r... | {{ProofWanted|It remains to be shown there are no other solutions. This can be achieved by brute force. For this page, it is then adequate to present a pseudocode program to illustrate it.}} | Solutions of Diophantine Equation x^4 + y^4 = z^2 + 1 for x = 239 | https://proofwiki.org/wiki/Solutions_of_Diophantine_Equation_x^4_+_y^4_=_z^2_+_1_for_x_=_239 | https://proofwiki.org/wiki/Solutions_of_Diophantine_Equation_x^4_+_y^4_=_z^2_+_1_for_x_=_239 | [
"Diophantine Equations",
"239"
] | [
"Definition:Indeterminate Equation",
"Definition:Diophantine Equation"
] | [] |
proofwiki-13251 | Smallest Numbers with 240 Divisors | The smallest integers with $240$ divisors are:
:$720 \, 720, 831 \, 600, 942 \, 480, 982 \, 800, 997 \, 920, \ldots$ | In the below, $\map {\sigma_0} n$ denotes the divisor count function of $n$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {720 \, 720}
| r = 240
| c = {{DCFLink|720,720|720 \, 720}}
}}
{{eqn | l = \map {\sigma_0} {831 \, 600}
| r = 240
| c = {{DCFLink|831,600|831 \, 600}}
}}
{{eqn | l = \map ... | The smallest [[Definition:Positive Integer|integers]] with $240$ [[Definition:Divisor of Integer|divisors]] are:
:$720 \, 720, 831 \, 600, 942 \, 480, 982 \, 800, 997 \, 920, \ldots$ | In the below, $\map {\sigma_0} n$ denotes the [[Definition:Divisor Count Function|divisor count function]] of $n$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {720 \, 720}
| r = 240
| c = {{DCFLink|720,720|720 \, 720}}
}}
{{eqn | l = \map {\sigma_0} {831 \, 600}
| r = 240
| c = {{DCFLink|8... | Smallest Numbers with 240 Divisors | https://proofwiki.org/wiki/Smallest_Numbers_with_240_Divisors | https://proofwiki.org/wiki/Smallest_Numbers_with_240_Divisors | [
"Divisor Count Function"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor Count Function"
] |
proofwiki-13252 | Sequence of 4 Consecutive Integers with Equal Number of Divisors | The following sequence of integers are sets of $4$ consecutive integers which all have the same number of divisors:
:$\map {\sigma_0} m = \map {\sigma_0} {m + 1} = \map {\sigma_0} {m + 2} = \map {\sigma_0} {m + 3}$
where $\map {\sigma_0} n$ denotes the divisor count function.
:$242, 243, 244, 245, 3655, 3656, 3657, 365... | {{begin-eqn}}
{{eqn | l = \map {\sigma_0} {242}
| r = 6
| c = {{DCFLink|242}}
}}
{{eqn | l = \map {\sigma_0} {243}
| r = 6
| c = {{DCFLink|243}}
}}
{{eqn | l = \map {\sigma_0} {244}
| r = 6
| c = {{DCFLink|244}}
}}
{{eqn | l = \map {\sigma_0} {245}
| r = 6
| c = {{DCFLink... | The following [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|integers]] are [[Definition:Set|sets]] of $4$ consecutive [[Definition:Positive Integer|integers]] which all have the same number of [[Definition:Divisor of Integer|divisors]]:
:$\map {\sigma_0} m = \map {\sigma_0} {m + 1} = \map {\... | {{begin-eqn}}
{{eqn | l = \map {\sigma_0} {242}
| r = 6
| c = {{DCFLink|242}}
}}
{{eqn | l = \map {\sigma_0} {243}
| r = 6
| c = {{DCFLink|243}}
}}
{{eqn | l = \map {\sigma_0} {244}
| r = 6
| c = {{DCFLink|244}}
}}
{{eqn | l = \map {\sigma_0} {245}
| r = 6
| c = {{DCFLink... | Sequence of 4 Consecutive Integers with Equal Number of Divisors | https://proofwiki.org/wiki/Sequence_of_4_Consecutive_Integers_with_Equal_Number_of_Divisors | https://proofwiki.org/wiki/Sequence_of_4_Consecutive_Integers_with_Equal_Number_of_Divisors | [
"Divisor Count Function"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor Count Function"
] | [] |
proofwiki-13253 | Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes/Sequence | The sequence of positive integers which are both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers begins:
:$65, 250, \ldots$ | {{begin-eqn}}
{{eqn | l = 65
| r = 4^3 + 1^3
| c =
}}
{{eqn | r = 8^2 + 1^2
| c =
}}
{{eqn | r = 7^2 + 4^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 250
| r = 5^3 + 5^3
| c =
}}
{{eqn | r = 15^2 + 5^2
| c =
}}
{{eqn | r = 13^2 + 9^2
| c =
}}
{{end-eqn}}
{{qed}} | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] which are both the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Square Number|square numbers]] in two [[Definition:Distinct|distinct]] ways and also the [[Definition:Integer Addition|sum]] of $2$ [[Definition:C... | {{begin-eqn}}
{{eqn | l = 65
| r = 4^3 + 1^3
| c =
}}
{{eqn | r = 8^2 + 1^2
| c =
}}
{{eqn | r = 7^2 + 4^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 250
| r = 5^3 + 5^3
| c =
}}
{{eqn | r = 15^2 + 5^2
| c =
}}
{{eqn | r = 13^2 + 9^2
| c =
}}
{{end-eqn}}
{{qed... | Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes/Sequence | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways_which_is_also_Sum_of_Cubes/Sequence | https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways_which_is_also_Sum_of_Cubes/Sequence | [
"Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct",
"Definition:Addition/Integers",
"Definition:Cube Number"
] | [] |
proofwiki-13254 | 78,557 is Sierpiński | $78 \, 557$ is a Sierpiński number of the second kind. | When considering $\bmod {36}$, every positive integer $n$ can be written in one of the forms:
:$2 k, 4 k + 1, 3 k + 1, 12 k + 11, 18 k + 15, 36 k + 27, 9 k + 3$
$\begin{array}{|c|c|}
\hline n \bmod {36} & \text { Form } \\
\hline 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34 & 2 k \\
\hline 1, 5, ... | $78 \, 557$ is a [[Definition:Sierpiński Number of the Second Kind|Sierpiński number of the second kind]]. | When considering $\bmod {36}$, every [[Definition:Positive Integer|positive integer]] $n$ can be written in one of the forms:
:$2 k, 4 k + 1, 3 k + 1, 12 k + 11, 18 k + 15, 36 k + 27, 9 k + 3$
$\begin{array}{|c|c|}
\hline n \bmod {36} & \text { Form } \\
\hline 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 3... | 78,557 is Sierpiński | https://proofwiki.org/wiki/78,557_is_Sierpiński | https://proofwiki.org/wiki/78,557_is_Sierpiński | [
"Sierpiński Numbers of the Second Kind",
"78,557"
] | [
"Definition:Sierpiński Number of the Second Kind"
] | [
"Definition:Positive/Integer",
"Fermat's Little Theorem",
"Congruence of Powers",
"Congruence of Product",
"Fermat's Little Theorem",
"Congruence of Powers",
"Congruence of Product",
"Congruence of Powers",
"Congruence of Product",
"Fermat's Little Theorem",
"Congruence of Powers",
"Congruence... |
proofwiki-13255 | Equivalence of Definitions of Balanced Prime | The following definitions of a balanced prime are equivalent: | {{begin-eqn}}
{{eqn | l = p_n
| r = \dfrac {p_{n - 1} + p_{n + 1} } 2
| c = {{Defof|Balanced Prime|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = 2 p_n
| r = p_{n - 1} + p_{n + 1}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = p_n - p_{n - 1}
| r = p_{n + 1} - p_n
| c = {{... | The following definitions of a [[Definition:Balanced Prime|balanced prime]] are [[Definition:Logical Equivalence|equivalent]]: | {{begin-eqn}}
{{eqn | l = p_n
| r = \dfrac {p_{n - 1} + p_{n + 1} } 2
| c = {{Defof|Balanced Prime|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = 2 p_n
| r = p_{n - 1} + p_{n + 1}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = p_n - p_{n - 1}
| r = p_{n + 1} - p_n
| c = {{... | Equivalence of Definitions of Balanced Prime | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Balanced_Prime | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Balanced_Prime | [
"Balanced Primes"
] | [
"Definition:Balanced Prime",
"Definition:Logical Equivalence"
] | [
"Definition:Prime Gap",
"Category:Balanced Primes"
] |
proofwiki-13256 | Filtered iff Directed in Dual Ordered Set | Let $\struct {S, \preceq_1}$ be an ordered set.
Let $\struct {S, \preceq_2}$ be a dual ordered set of $\struct {S, \preceq_1}$
Let $X \subseteq S$.
Then:
:$X$ is filtered in $\struct {S, \preceq_1}$
{{iff}}:
:$X$ is directed in $\struct {S, \preceq_2}$ | By Dual of Dual Ordering:
:$\struct {S, \preceq_1}$ is the dual of $\struct {S, \preceq_2}$.
Thus by Directed iff Filtered in Dual Ordered Set:
:$X$ is filtered in $\struct {S, \preceq_1}$
{{iff}}:
:$X$ is directed in $\struct {S, \preceq_2}$.
{{qed}} | Let $\struct {S, \preceq_1}$ be an [[Definition:Ordered Set|ordered set]].
Let $\struct {S, \preceq_2}$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $\struct {S, \preceq_1}$
Let $X \subseteq S$.
Then:
:$X$ is [[Definition:Filtered Subset|filtered]] in $\struct {S, \preceq_1}$
{{iff}}:
:$X$ is [[Definiti... | By [[Dual of Dual Ordering]]:
:$\struct {S, \preceq_1}$ is the [[Definition:Dual Ordered Set|dual]] of $\struct {S, \preceq_2}$.
Thus by [[Directed iff Filtered in Dual Ordered Set]]:
:$X$ is [[Definition:Filtered Subset|filtered]] in $\struct {S, \preceq_1}$
{{iff}}:
:$X$ is [[Definition:Directed Subset|directed]] in... | Filtered iff Directed in Dual Ordered Set | https://proofwiki.org/wiki/Filtered_iff_Directed_in_Dual_Ordered_Set | https://proofwiki.org/wiki/Filtered_iff_Directed_in_Dual_Ordered_Set | [
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Filtered Subset",
"Definition:Directed Subset"
] | [
"Dual of Dual Ordering",
"Definition:Dual Ordering/Dual Ordered Set",
"Directed iff Filtered in Dual Ordered Set",
"Definition:Filtered Subset",
"Definition:Directed Subset"
] |
proofwiki-13257 | Filters equal Ideals in Dual Ordered Set | Let $L_1 = \struct {S, \preceq_1}$ be an ordered set.
Let $L_2 = \struct {S, \preceq_2}$ be a dual ordered set of $L_1$.
Then $\map {\mathit {Filt} } {L_1} = \map {\mathit {Ids} } {L_2}$
where
:$\map {\mathit {Filt} } {L_1}$ denotes the set of all filters of $L_1$,
:$\map {\mathit {Ids} } {L_2}$ denotes the set of all ... | Let $x$ be a set.
By definition of $\mathit {Filt}$:
:$x \in \map {\mathit {Filt} } {L_1} \iff x$ is a filter on $L_1$.
By Filter is Ideal in Dual Ordered Set:
:$x \in \map {\mathit {Filt} } {L_1} \iff x$ is an ideal in $L_2$.
By definition of $\mathit{Ids}$:
:$x \in \map {\mathit {Filt} } {L_1} \iff x \in \map {\mathi... | Let $L_1 = \struct {S, \preceq_1}$ be an [[Definition:Ordered Set|ordered set]].
Let $L_2 = \struct {S, \preceq_2}$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $L_1$.
Then $\map {\mathit {Filt} } {L_1} = \map {\mathit {Ids} } {L_2}$
where
:$\map {\mathit {Filt} } {L_1}$ denotes the [[Definition:Set of ... | Let $x$ be a [[Definition:Set|set]].
By definition of $\mathit {Filt}$:
:$x \in \map {\mathit {Filt} } {L_1} \iff x$ is a [[Definition:Filter|filter]] on $L_1$.
By [[Filter is Ideal in Dual Ordered Set]]:
:$x \in \map {\mathit {Filt} } {L_1} \iff x$ is an [[Definition:Ideal (Order Theory)|ideal]] in $L_2$.
By defini... | Filters equal Ideals in Dual Ordered Set | https://proofwiki.org/wiki/Filters_equal_Ideals_in_Dual_Ordered_Set | https://proofwiki.org/wiki/Filters_equal_Ideals_in_Dual_Ordered_Set | [
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Set of Sets",
"Definition:Filter",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)"
] | [
"Definition:Set",
"Definition:Filter",
"Filter is Ideal in Dual Ordered Set",
"Definition:Ideal (Order Theory)",
"Definition:Set Equality"
] |
proofwiki-13258 | Pépin's Test | Let $F_n = 2^{2^n} + 1$ be a Fermat number.
Then $F_n$ is prime {{iff}}:
:$3^{\paren {F_n - 1} / 2} \equiv -1 \pmod {F_n}$ | === Sufficient Condition ===
Let this congruence hold:
:$3^{\paren {F_n - 1} / 2} \equiv -1 \pmod {F_n}$
Then:
:$3^{F_n - 1} \equiv 1 \pmod {F_n}$
Thus the multiplicative order of $3$ modulo $F_n$ is a divisor of $F_n - 1 = 2^{2^n}$.
This is a power of $2$.
On the other hand, the multiplicative order of $3$ modulo $F_n... | Let $F_n = 2^{2^n} + 1$ be a [[Definition:Fermat Number|Fermat number]].
Then $F_n$ is [[Definition:Prime Number|prime]] {{iff}}:
:$3^{\paren {F_n - 1} / 2} \equiv -1 \pmod {F_n}$ | === Sufficient Condition ===
Let this [[Definition:Congruence Modulo Integer|congruence]] hold:
:$3^{\paren {F_n - 1} / 2} \equiv -1 \pmod {F_n}$
Then:
:$3^{F_n - 1} \equiv 1 \pmod {F_n}$
Thus the [[Definition:Multiplicative Order of Integer|multiplicative order of $3$ modulo $F_n$]] is a [[Definition:Divisor of Int... | Pépin's Test | https://proofwiki.org/wiki/Pépin's_Test | https://proofwiki.org/wiki/Pépin's_Test | [
"Fermat Numbers"
] | [
"Definition:Fermat Number",
"Definition:Prime Number"
] | [
"Definition:Congruence (Number Theory)/Integers",
"Definition:Multiplicative Order of Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Multiplicative Order of Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Coprime/Integers",
"Definition... |
proofwiki-13259 | Triangular Fermat Number | The only one Fermat number which is triangular is $3$. | Let $F_n$ be a Fermat number which is triangular.
We use the weaker form of Divisor of Fermat Number:
:Every divisor of $F_n$ is of the form $k \times 2^{n + 1} + 1$
as this formulation is valid for all $n \in \N$.
From Closed Form for Triangular Numbers:
:$F_n = \dfrac {m \paren {m + 1} } 2$
Either $m$ or $m + 1$ is e... | The only one [[Definition:Fermat Number|Fermat number]] which is [[Definition:Triangular Number|triangular]] is $3$. | Let $F_n$ be a [[Definition:Fermat Number|Fermat number]] which is [[Definition:Triangular Number|triangular]].
We use the weaker form of [[Divisor of Fermat Number]]:
:Every [[Definition:Divisor of Integer|divisor]] of $F_n$ is of the form $k \times 2^{n + 1} + 1$
as this formulation is valid for all $n \in \N$.
F... | Triangular Fermat Number | https://proofwiki.org/wiki/Triangular_Fermat_Number | https://proofwiki.org/wiki/Triangular_Fermat_Number | [
"Fermat Numbers",
"Triangular Numbers"
] | [
"Definition:Fermat Number",
"Definition:Triangular Number"
] | [
"Definition:Fermat Number",
"Definition:Triangular Number",
"Divisor of Fermat Number",
"Definition:Divisor (Algebra)/Integer",
"Closed Form for Triangular Numbers",
"Definition:Even Integer",
"Definition:Even Integer",
"Divisor of Fermat Number",
"Divisor of Fermat Number",
"Definition:Odd Intege... |
proofwiki-13260 | Fermat Number is not Square | There exist no Fermat numbers which are square. | Recall the definition of Fermat numbers:
:$F_n = 2^{(2^n)}+1$, where $n = 0, 1, 2, \ldots$
=== Marginal Case ===
$F_0 = 3$ is not a square.
=== General Case ===
It will be shown that Fermat numbers lie between $2$ consecutive squares, thus cannot be itself a square:
{{begin-eqn}}
{{eqn | l = \left({2^{ \left({2^{n-1} }... | There exist no [[Definition:Fermat Number|Fermat numbers]] which are [[Definition:Square Number|square]]. | Recall the definition of [[Definition:Fermat Number|Fermat numbers]]:
:$F_n = 2^{(2^n)}+1$, where $n = 0, 1, 2, \ldots$
=== Marginal Case ===
$F_0 = 3$ is not a [[Definition:Square Number|square]].
=== General Case ===
It will be shown that [[Definition:Fermat Number|Fermat numbers]] lie between $2$ consecutive ... | Fermat Number is not Square/Proof 2 | https://proofwiki.org/wiki/Fermat_Number_is_not_Square | https://proofwiki.org/wiki/Fermat_Number_is_not_Square/Proof_2 | [
"Fermat Numbers",
"Square Numbers",
"Fermat Number is not Square"
] | [
"Definition:Fermat Number",
"Definition:Square Number"
] | [
"Definition:Fermat Number",
"Definition:Square Number",
"Definition:Fermat Number",
"Definition:Square Number",
"Definition:Square Number",
"Index Laws/Product of Indices",
"Completing the Square"
] |
proofwiki-13261 | Fermat Number is not Cube | There exist no Fermat numbers which are cubes. | Direct consequence of Fermat Number is not Perfect Power.
{{qed}} | There exist no [[Definition:Fermat Number|Fermat numbers]] which are [[Definition:Cube Number|cubes]]. | Direct consequence of [[Fermat Number is not Perfect Power]].
{{qed}} | Fermat Number is not Cube/Proof 1 | https://proofwiki.org/wiki/Fermat_Number_is_not_Cube | https://proofwiki.org/wiki/Fermat_Number_is_not_Cube/Proof_1 | [
"Fermat Numbers",
"Cube Numbers",
"Fermat Number is not Cube"
] | [
"Definition:Fermat Number",
"Definition:Cube Number"
] | [
"Fermat Number is not Perfect Power"
] |
proofwiki-13262 | Construction of Regular 257-Gon | It is possible to construct a regular polygon with $257$ sides using a compass and straightedge construction. | 257 is a Fermat prime.
From Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime it is known that this construction is possible.
{{qed}} | It is possible to construct a [[Definition:Regular Polygon|regular polygon]] with $257$ [[Definition:Side of Polygon|sides]] using a [[Definition:Compass and Straightedge Construction|compass and straightedge construction]]. | [[257]] is a [[Definition:Fermat Prime|Fermat prime]].
From [[Construction of Regular Prime p-Gon Exists iff p is Fermat Prime|Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime]] it is known that this construction is possible.
{{qed}} | Construction of Regular 257-Gon | https://proofwiki.org/wiki/Construction_of_Regular_257-Gon | https://proofwiki.org/wiki/Construction_of_Regular_257-Gon | [
"Compass and Straightedge Constructions",
"Regular Polygons",
"257"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Definition:Compass and Straightedge Construction"
] | [
"257",
"Definition:Fermat Prime",
"Construction of Regular Prime p-Gon Exists iff p is Fermat Prime"
] |
proofwiki-13263 | Ideals equal Filters in Dual Ordered Set | Let $L_1 = \struct {S, \preceq_1}$ be an ordered set.
Let $L_2 = \struct {S, \preceq_2}$ be a dual ordered set of $L_1$.
Then $\map {\mathit {Ids} } {L_1} = \map {\mathit {Filt} } {L_2}$
where
:$\map {\mathit {Filt} } {L_2}$ denotes the set of all filters of $L_2$
:$\map {\mathit {Ids} } {L_1}$ denotes the set of all i... | Let $x$ be a set.
By definition of $\mathit {Filt}$:
:$x \in \map {\mathit {Filt} } {L_2} \iff x$ is a filter on $L_2$.
By Ideal is Filter in Dual Ordered Set:
:$x \in \map {\mathit {Filt} } {L_2} \iff x$ is an ideal in $L_1$.
By definition of $\mathit{Ids}$:
:$x \in \map {\mathit {Filt} } {L_2} \iff x \in \map {\mathi... | Let $L_1 = \struct {S, \preceq_1}$ be an [[Definition:Ordered Set|ordered set]].
Let $L_2 = \struct {S, \preceq_2}$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $L_1$.
Then $\map {\mathit {Ids} } {L_1} = \map {\mathit {Filt} } {L_2}$
where
:$\map {\mathit {Filt} } {L_2}$ denotes the [[Definition:Set of ... | Let $x$ be a [[Definition:Set|set]].
By definition of $\mathit {Filt}$:
:$x \in \map {\mathit {Filt} } {L_2} \iff x$ is a [[Definition:Filter|filter]] on $L_2$.
By [[Ideal is Filter in Dual Ordered Set]]:
:$x \in \map {\mathit {Filt} } {L_2} \iff x$ is an [[Definition:Ideal (Order Theory)|ideal]] in $L_1$.
By defini... | Ideals equal Filters in Dual Ordered Set | https://proofwiki.org/wiki/Ideals_equal_Filters_in_Dual_Ordered_Set | https://proofwiki.org/wiki/Ideals_equal_Filters_in_Dual_Ordered_Set | [
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Set of Sets",
"Definition:Filter",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)"
] | [
"Definition:Set",
"Definition:Filter",
"Ideal is Filter in Dual Ordered Set",
"Definition:Ideal (Order Theory)",
"Definition:Set Equality"
] |
proofwiki-13264 | Power Set is Filter in Lattice of Power Set | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be a inclusion lattice of power set of $X$.
Then $\powerset X$ is a filter on $L$. | === Filtered ===
By Set is Element of its Power Set:
:$X \in \powerset X$
Thus by definition:
:$\powerset X$ is a non-empty set.
Let $x, y \in \powerset X$.
By Intersection is Subset:
:$x \cap y \subseteq x$ and $x \cap y \subseteq y$
By Subset Relation is Transitive:
:$x \cap y \in \powerset X$
Thus
:$\exists z \in \p... | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be a [[Definition:Inclusion Ordered Set|inclusion]] [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]] of $X$.
Then $\powerset X$ is a [[Definition:Filter|filter]] on $L$. | === Filtered ===
By [[Set is Element of its Power Set]]:
:$X \in \powerset X$
Thus by definition:
:$\powerset X$ is a [[Definition:Non-Empty Set|non-empty set]].
Let $x, y \in \powerset X$.
By [[Intersection is Subset]]:
:$x \cap y \subseteq x$ and $x \cap y \subseteq y$
By [[Subset Relation is Transitive]]:
:$x \... | Power Set is Filter in Lattice of Power Set | https://proofwiki.org/wiki/Power_Set_is_Filter_in_Lattice_of_Power_Set | https://proofwiki.org/wiki/Power_Set_is_Filter_in_Lattice_of_Power_Set | [
"Power Set"
] | [
"Definition:Set",
"Definition:Inclusion Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Filter"
] | [
"Set is Element of its Power Set",
"Definition:Non-Empty Set",
"Intersection is Subset",
"Subset Relation is Transitive"
] |
proofwiki-13265 | Triplet in Arithmetic Sequence with equal Divisor Sum | The smallest triple of integers in arithmetic sequence which have the same divisor sum is:
:$\map {\sigma_1} {267} = \map {\sigma_1} {295} = \map {\sigma_1} {323} = 360$ | We have that:
{{begin-eqn}}
{{eqn | l = 295 - 267
| r = 28
| c =
}}
{{eqn | l = 323 - 295
| r = 28
| c =
}}
{{end-eqn}}
demonstrating that $267, 295, 323$ are in arithmetic sequence with common difference $28$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {267}
| r = 360
| c = {... | The smallest [[Definition:Ordered Triple|triple]] of [[Definition:Positive Integer|integers]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] which have the same [[Definition:Divisor Sum Function|divisor sum]] is:
:$\map {\sigma_1} {267} = \map {\sigma_1} {295} = \map {\sigma_1} {323} = 360$ | We have that:
{{begin-eqn}}
{{eqn | l = 295 - 267
| r = 28
| c =
}}
{{eqn | l = 323 - 295
| r = 28
| c =
}}
{{end-eqn}}
demonstrating that $267, 295, 323$ are in [[Definition:Arithmetic Sequence|arithmetic sequence]] with [[Definition:Common Difference|common difference]] $28$.
Then:
{{be... | Triplet in Arithmetic Sequence with equal Divisor Sum | https://proofwiki.org/wiki/Triplet_in_Arithmetic_Sequence_with_equal_Divisor_Sum | https://proofwiki.org/wiki/Triplet_in_Arithmetic_Sequence_with_equal_Divisor_Sum | [
"Divisor Sum Function",
"Arithmetic Sequences",
"267",
"295",
"323"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Positive/Integer",
"Definition:Arithmetic Sequence",
"Definition:Divisor Sum Function"
] | [
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Difference"
] |
proofwiki-13266 | Singleton of Set is Filter in Lattice of Power Set | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be an inclusion lattice of power set of $X$.
Then $\set X$ is a filter on $L$. | By Singleton is Directed and Filtered Subset:
:$\set X$ is filtered.
We will prove that
:$\set X$ is an upper section.
Let $x \in \set X$, $y \in \powerset X$ such that:
:$x \subseteq y$
By definition of singleton:
:$x = X$
By definition of power set:
:$y \subseteq X$
By definition of set equality:
:$y = X$
Thus:
:$y \... | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be an [[Definition:Inclusion Ordered Set|inclusion]] [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]] of $X$.
Then $\set X$ is a [[Definition:Filter|filter]] on $L$. | By [[Singleton is Directed and Filtered Subset]]:
:$\set X$ is [[Definition:Filtered Subset|filtered]].
We will prove that
:$\set X$ is an [[Definition:Upper Section|upper section]].
Let $x \in \set X$, $y \in \powerset X$ such that:
:$x \subseteq y$
By definition of [[Definition:Singleton|singleton]]:
:$x = X$
By ... | Singleton of Set is Filter in Lattice of Power Set | https://proofwiki.org/wiki/Singleton_of_Set_is_Filter_in_Lattice_of_Power_Set | https://proofwiki.org/wiki/Singleton_of_Set_is_Filter_in_Lattice_of_Power_Set | [
"Power Set"
] | [
"Definition:Set",
"Definition:Inclusion Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Filter"
] | [
"Singleton is Directed and Filtered Subset",
"Definition:Filtered Subset",
"Definition:Upper Section",
"Definition:Singleton",
"Definition:Power Set",
"Definition:Set Equality",
"Definition:Filter in Ordered Set",
"Definition:Filter"
] |
proofwiki-13267 | Existence of Arbitrarily Long Aliquot Sequences | It is possible to construct an aliquot sequence of arbitrary length which is monotonically increasing. | Let $p_i$ be the $i$th prime.
Construct the sequence $\sequence {t_i}_{t \mathop = 1}^\infty$ of natural numbers $t_i$ recursively as follows:
:$t_1 = 2$
:$t_{i + 1} = \map \phi {p_i^{t_i + 1} } - 1 = p_i^{t_i} \paren {p_i - 1} - 1$ for $i \ge 1$
We claim that this sequence has the property:
:$(1): \quad p_i^{t_i + 1} ... | It is possible to construct an [[Definition:Aliquot Sequence|aliquot sequence]] of arbitrary [[Definition:Length of Sequence|length]] which is [[Definition:Monotone Sequence|monotonically]] [[Definition:Increasing Sequence|increasing]]. | Let $p_i$ be the $i$th [[Definition:Prime Number|prime]].
Construct the sequence $\sequence {t_i}_{t \mathop = 1}^\infty$ of [[Definition:Natural Numbers|natural numbers]] $t_i$ recursively as follows:
:$t_1 = 2$
:$t_{i + 1} = \map \phi {p_i^{t_i + 1} } - 1 = p_i^{t_i} \paren {p_i - 1} - 1$ for $i \ge 1$
We claim tha... | Existence of Arbitrarily Long Aliquot Sequences | https://proofwiki.org/wiki/Existence_of_Arbitrarily_Long_Aliquot_Sequences | https://proofwiki.org/wiki/Existence_of_Arbitrarily_Long_Aliquot_Sequences | [
"Aliquot Sequences"
] | [
"Definition:Aliquot Sequence",
"Definition:Length of Sequence",
"Definition:Monotone (Order Theory)/Sequence",
"Definition:Increasing/Sequence"
] | [
"Definition:Prime Number",
"Definition:Natural Numbers",
"Divisor Sum of Power of Prime",
"Euler's Theorem (Number Theory)",
"Definition:Even Integer",
"Definition:Composite Number",
"Definition:Odd Integer",
"Bertrand-Chebyshev Theorem",
"Definition:Coprime/Integers",
"Euclid's Lemma",
"Definit... |
proofwiki-13268 | Consecutive Triple of Repeated Digit-Products | The triplet of integers $281, 282, 283$ have the property that if their digits are multiplied, and the process repeated on the result until only $1$ digit remains, that final digit is the same for all three, that is, $6$.
There does not exist an set of four consecutive integers which also all end up at the same single ... | {{begin-eqn}}
{{eqn | ll= 281:
| l = 2 \times 8 \times 1
| r = 16
| c =
}}
{{eqn | l = 1 \times 6
| r = 6
| c =
}}
{{eqn | ll= 282:
| l = 2 \times 8 \times 2
| r = 32
| c =
}}
{{eqn | l = 3 \times 2
| r = 6
| c =
}}
{{eqn | ll= 283:
| l = 2 \times 8 ... | The [[Definition:Ordered Triple|triplet]] of [[Definition:Integer|integers]] $281, 282, 283$ have the property that if their [[Definition:Digit|digits]] are [[Definition:Integer Multiplication|multiplied]], and the process repeated on the result until only $1$ [[Definition:Digit|digit]] remains, that final [[Definition... | {{begin-eqn}}
{{eqn | ll= 281:
| l = 2 \times 8 \times 1
| r = 16
| c =
}}
{{eqn | l = 1 \times 6
| r = 6
| c =
}}
{{eqn | ll= 282:
| l = 2 \times 8 \times 2
| r = 32
| c =
}}
{{eqn | l = 3 \times 2
| r = 6
| c =
}}
{{eqn | ll= 283:
| l = 2 \times 8 ... | Consecutive Triple of Repeated Digit-Products | https://proofwiki.org/wiki/Consecutive_Triple_of_Repeated_Digit-Products | https://proofwiki.org/wiki/Consecutive_Triple_of_Repeated_Digit-Products | [
"281",
"282",
"283",
"Recreational Mathematics"
] | [
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Integer",
"Definition:Digit",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Digit",
"Definition:Ordered Tuple as Ordered Set/Ordered Quadruple",
"Definition:Integer",
"Definition:Digit"
] | [] |
proofwiki-13269 | Consecutive Powerful Numbers | The following pairs are of consecutive positive integers both of which are powerful:
:$\left({8, 9}\right), \left({288, 289}\right), \left({675, 676}\right), \left({9800, 9801}\right), \left({332 \, 928, 332 \, 929}\right), \ldots$
{{OEIS|A060355}} | {{begin-eqn}}
{{eqn | l = 8
| r = 2^3
}}
{{eqn | l = 9
| r = 3^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 288
| r = 2^5 \times 3^2
}}
{{eqn | l = 289
| r = 17^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 675
| r = 3^3 \times 5^2
}}
{{eqn | l = 676
| r = 2^2 \times 13^2
}}
{{end-eqn}}
{... | The following [[Definition:Ordered Pair|pairs]] are of consecutive [[Definition:Positive Integer|positive integers]] both of which are [[Definition:Powerful Number|powerful]]:
:$\left({8, 9}\right), \left({288, 289}\right), \left({675, 676}\right), \left({9800, 9801}\right), \left({332 \, 928, 332 \, 929}\right), \ldot... | {{begin-eqn}}
{{eqn | l = 8
| r = 2^3
}}
{{eqn | l = 9
| r = 3^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 288
| r = 2^5 \times 3^2
}}
{{eqn | l = 289
| r = 17^2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 675
| r = 3^3 \times 5^2
}}
{{eqn | l = 676
| r = 2^2 \times 13^2
}}
{{end-eqn... | Consecutive Powerful Numbers | https://proofwiki.org/wiki/Consecutive_Powerful_Numbers | https://proofwiki.org/wiki/Consecutive_Powerful_Numbers | [
"Powerful Numbers"
] | [
"Definition:Ordered Pair",
"Definition:Positive/Integer",
"Definition:Powerful Number"
] | [] |
proofwiki-13270 | Smallest Multiple of 9 with all Digits Even | $288$ is the smallest integer multiple of $9$ all of whose digits are even. | By the brute force technique:
{{begin-eqn}}
{{eqn | l = 1 \times 9
| r = 9
}}
{{eqn | l = 2 \times 9
| r = 18
}}
{{eqn | l = 3 \times 9
| r = 27
}}
{{eqn | l = 4 \times 9
| r = 36
}}
{{eqn | l = 5 \times 9
| r = 45
}}
{{eqn | l = 6 \times 9
| r = 54
}}
{{eqn | l = 7 \times 9
| ... | $288$ is the smallest [[Definition:Integer Multiple|integer multiple of $9$]] all of whose [[Definition:Digit|digits]] are [[Definition:Even Integer|even]]. | By the brute force technique:
{{begin-eqn}}
{{eqn | l = 1 \times 9
| r = 9
}}
{{eqn | l = 2 \times 9
| r = 18
}}
{{eqn | l = 3 \times 9
| r = 27
}}
{{eqn | l = 4 \times 9
| r = 36
}}
{{eqn | l = 5 \times 9
| r = 45
}}
{{eqn | l = 6 \times 9
| r = 54
}}
{{eqn | l = 7 \times 9
|... | Smallest Multiple of 9 with all Digits Even/Proof 1 | https://proofwiki.org/wiki/Smallest_Multiple_of_9_with_all_Digits_Even | https://proofwiki.org/wiki/Smallest_Multiple_of_9_with_all_Digits_Even/Proof_1 | [
"Recreational Mathematics",
"9",
"288",
"Classic Problems",
"Smallest Multiple of 9 with all Digits Even"
] | [
"Definition:Integral Multiple/Real Numbers",
"Definition:Digit",
"Definition:Even Integer"
] | [
"Definition:Integral Multiple/Real Numbers",
"Definition:Odd Integer",
"Definition:Digit"
] |
proofwiki-13271 | Smallest Multiple of 9 with all Digits Even | $288$ is the smallest integer multiple of $9$ all of whose digits are even. | Let $n$ be the smallest integer multiple of $9$ all of whose digits are even.
From Divisibility by 9, the digits of $n$ must add to an integer multiple of $9$.
But from Sum of Even Integers is Even, the digits of $n$ must add to an even integer multiple of $9$: $18, 36, 54$, etc.
There is only $1$ integer multiple of $... | $288$ is the smallest [[Definition:Integer Multiple|integer multiple of $9$]] all of whose [[Definition:Digit|digits]] are [[Definition:Even Integer|even]]. | Let $n$ be the smallest [[Definition:Integer Multiple|integer multiple of $9$]] all of whose [[Definition:Digit|digits]] are [[Definition:Even Integer|even]].
From [[Divisibility by 9]], the [[Definition:Digit|digits]] of $n$ must add to an [[Definition:Integer Multiple|integer multiple of $9$]].
But from [[Sum of Ev... | Smallest Multiple of 9 with all Digits Even/Proof 2 | https://proofwiki.org/wiki/Smallest_Multiple_of_9_with_all_Digits_Even | https://proofwiki.org/wiki/Smallest_Multiple_of_9_with_all_Digits_Even/Proof_2 | [
"Recreational Mathematics",
"9",
"288",
"Classic Problems",
"Smallest Multiple of 9 with all Digits Even"
] | [
"Definition:Integral Multiple/Real Numbers",
"Definition:Digit",
"Definition:Even Integer"
] | [
"Definition:Integral Multiple/Real Numbers",
"Definition:Digit",
"Definition:Even Integer",
"Divisibility by 9",
"Definition:Digit",
"Definition:Integral Multiple/Real Numbers",
"Sum of Even Integers is Even",
"Definition:Digit",
"Definition:Even Integer",
"Definition:Integral Multiple/Real Number... |
proofwiki-13272 | Kaprekar's Symmetry | Let $n$ be a Kaprekar number with $D$ digits.
Then $10^D - n$ is also a Kaprekar number. | Since $n$ is a Kaprekar number of $D$ digits:
:$\begin {cases} n^2 = a \times 10^D + b \\ n = a + b \end {cases}$
for some positive integers $a$ and $b$, $b < 10^D$.
Hence:
{{begin-eqn}}
{{eqn | l = \paren {10^D - n}^2
| r = 10^{2 D} - 2 n \times 10^D + n^2
}}
{{eqn | r = 10^{2 D} - 2 \paren {a + b} 10^D + a \tim... | Let $n$ be a [[Definition:Kaprekar Number|Kaprekar number]] with $D$ [[Definition:Digit|digits]].
Then $10^D - n$ is also a [[Definition:Kaprekar Number|Kaprekar number]]. | Since $n$ is a [[Definition:Kaprekar Number|Kaprekar number]] of $D$ [[Definition:Digit|digits]]:
:$\begin {cases} n^2 = a \times 10^D + b \\ n = a + b \end {cases}$
for some [[Definition:Positive Integer|positive integers]] $a$ and $b$, $b < 10^D$.
Hence:
{{begin-eqn}}
{{eqn | l = \paren {10^D - n}^2
| r = 10... | Kaprekar's Symmetry | https://proofwiki.org/wiki/Kaprekar's_Symmetry | https://proofwiki.org/wiki/Kaprekar's_Symmetry | [
"Kaprekar Numbers"
] | [
"Definition:Kaprekar Number",
"Definition:Digit",
"Definition:Kaprekar Number"
] | [
"Definition:Kaprekar Number",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Square Number",
"Definition:Positive/Integer",
"Definition:Contradiction",
"Definition:Kaprekar Number",
"Definition:Digit"
] |
proofwiki-13273 | Difference between Kaprekar Number and Square is Multiple of Repunit | Let $n$ be a Kaprekar number of $m$ digits.
Then:
: $n^2 - n = k R_m$
where:
:$R_m$ is the $m$-digit repunit
:$k$ is an integer. | Since $n$ is a Kaprekar number of $m$ digits:
:$\begin {cases} n^2 = a \times 10^m + b \\ n = a + b \end {cases}$
for some positive integers $a$ and $b$.
Hence:
{{begin-eqn}}
{{eqn | l = n^2 - n
| r = a \times 10^m + b - a - b
}}
{{eqn | r = a \paren {10^m - 1}
}}
{{eqn | r = 9 a R_m
}}
{{end-eqn}}
{{qed}} | Let $n$ be a [[Definition:Kaprekar Number|Kaprekar number]] of $m$ [[Definition:Digit|digits]].
Then:
: $n^2 - n = k R_m$
where:
:$R_m$ is the [[Definition:Repunit|$m$-digit repunit]]
:$k$ is an [[Definition:Integer|integer]]. | Since $n$ is a [[Definition:Kaprekar Number|Kaprekar number]] of $m$ [[Definition:Digit|digits]]:
:$\begin {cases} n^2 = a \times 10^m + b \\ n = a + b \end {cases}$
for some [[Definition:Positive Integer|positive integers]] $a$ and $b$.
Hence:
{{begin-eqn}}
{{eqn | l = n^2 - n
| r = a \times 10^m + b - a - b
... | Difference between Kaprekar Number and Square is Multiple of Repunit | https://proofwiki.org/wiki/Difference_between_Kaprekar_Number_and_Square_is_Multiple_of_Repunit | https://proofwiki.org/wiki/Difference_between_Kaprekar_Number_and_Square_is_Multiple_of_Repunit | [
"Kaprekar Numbers"
] | [
"Definition:Kaprekar Number",
"Definition:Digit",
"Definition:Repunit",
"Definition:Integer"
] | [
"Definition:Kaprekar Number",
"Definition:Digit",
"Definition:Positive/Integer"
] |
proofwiki-13274 | Filters of Lattice of Power Set form Bounded Above Ordered Set | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be an inclusion lattice of power set of $X$.
Let $F = \struct {\map {\operatorname{Filt} } L, \subseteq}$ be an inclusion ordered set,
where $\map {\operatorname{Filt} } L$ denotes the set of all filters on $L$.
Then $F$ is bounded above and $\top... | By Power Set is Filter in Lattice of Power Set:
:$\powerset X$ is a filter on $L$.
Let $A \in \map {\operatorname{Filt} } L$.
Thus by definition of filter:
:$A \subseteq \powerset X$
Thus by definitions:
:$F$ is bounded above and $\top_F = \powerset X$
{{qed}} | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be an [[Definition:Inclusion Ordered Set|inclusion]] [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]] of $X$.
Let $F = \struct {\map {\operatorname{Filt} } L, \subseteq}$ be an [[Definition:I... | By [[Power Set is Filter in Lattice of Power Set]]:
:$\powerset X$ is a [[Definition:Filter|filter]] on $L$.
Let $A \in \map {\operatorname{Filt} } L$.
Thus by definition of [[Definition:Filter|filter]]:
:$A \subseteq \powerset X$
Thus by definitions:
:$F$ is [[Definition:Bounded Above Set|bounded above]] and $\top_... | Filters of Lattice of Power Set form Bounded Above Ordered Set | https://proofwiki.org/wiki/Filters_of_Lattice_of_Power_Set_form_Bounded_Above_Ordered_Set | https://proofwiki.org/wiki/Filters_of_Lattice_of_Power_Set_form_Bounded_Above_Ordered_Set | [
"Power Set"
] | [
"Definition:Set",
"Definition:Inclusion Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Inclusion Ordered Set",
"Definition:Set of Sets",
"Definition:Filter",
"Definition:Bounded Above Set",
"Definition:Greatest Element"
] | [
"Power Set is Filter in Lattice of Power Set",
"Definition:Filter",
"Definition:Filter",
"Definition:Bounded Above Set"
] |
proofwiki-13275 | Filters of Lattice of Power Set form Bounded Below Ordered Set | Let $X$ be a set.
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be an inclusion lattice of power set of $X$.
Let $F = \struct {\map {\operatorname {Filt} } L, \subseteq}$ be an inclusion ordered set,
where $\map {\operatorname {Filt} } L$ denotes the set of all filters on $L$.
Then $F$ is bounded below and $\b... | By Singleton of Set is Filter in Lattice of Power Set:
:$\set X$ is a filter on $L$.
Let $A \in \map {\operatorname {Filt} } L$.
By definition of non-empty set:
:$\exists x: x \in A$
By definition of power set:
:$x \subseteq X$
By definition of upper section:
:$X \in A$
Thus by definitions of singleton and subset:
:$\s... | Let $X$ be a [[Definition:Set|set]].
Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be an [[Definition:Inclusion Ordered Set|inclusion]] [[Definition:Lattice (Order Theory)|lattice]] of [[Definition:Power Set|power set]] of $X$.
Let $F = \struct {\map {\operatorname {Filt} } L, \subseteq}$ be an [[Definition:... | By [[Singleton of Set is Filter in Lattice of Power Set]]:
:$\set X$ is a [[Definition:Filter|filter]] on $L$.
Let $A \in \map {\operatorname {Filt} } L$.
By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\exists x: x \in A$
By definition of [[Definition:Power Set|power set]]:
:$x \subseteq X$
By defin... | Filters of Lattice of Power Set form Bounded Below Ordered Set | https://proofwiki.org/wiki/Filters_of_Lattice_of_Power_Set_form_Bounded_Below_Ordered_Set | https://proofwiki.org/wiki/Filters_of_Lattice_of_Power_Set_form_Bounded_Below_Ordered_Set | [
"Power Set"
] | [
"Definition:Set",
"Definition:Inclusion Ordered Set",
"Definition:Lattice (Order Theory)",
"Definition:Power Set",
"Definition:Inclusion Ordered Set",
"Definition:Set of Sets",
"Definition:Filter",
"Definition:Bounded Below Set",
"Definition:Smallest Element"
] | [
"Singleton of Set is Filter in Lattice of Power Set",
"Definition:Filter",
"Definition:Non-Empty Set",
"Definition:Power Set",
"Definition:Upper Section",
"Definition:Singleton",
"Definition:Subset",
"Definition:Bounded Below Set"
] |
proofwiki-13276 | Palindromic Primes in Base 10 and Base 2 | The following $n \in \Z$ are prime numbers which are palindromic in both decimal and binary:
:$3, 5, 7, 313, 7 \, 284 \, 717 \, 174 \, 827, 390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093, \ldots$
{{OEIS|A046472}}
It is not known whether there are any more. | :{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n_{10}$
! align="center" style = "padding: 2px 10px" | $n_2$
|-
| align="right" style = "padding: 2px 10px" | $3$
| align="right" style = "padding: 2px 10px" | $11$
|-
| align="right" style = "padding: 2px 10px" | $5$
| align="right" style = "padding: ... | The following $n \in \Z$ are [[Definition:Prime Number|prime numbers]] which are [[Definition:Palindromic Number|palindromic]] in both [[Definition:Decimal Notation|decimal]] and [[Definition:Binary Notation|binary]]:
:$3, 5, 7, 313, 7 \, 284 \, 717 \, 174 \, 827, 390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \,... | :{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n_{10}$
! align="center" style = "padding: 2px 10px" | $n_2$
|-
| align="right" style = "padding: 2px 10px" | $3$
| align="right" style = "padding: 2px 10px" | $11$
|-
| align="right" style = "padding: 2px 10px" | $5$
| align="right" style = "padding: ... | Palindromic Primes in Base 10 and Base 2 | https://proofwiki.org/wiki/Palindromic_Primes_in_Base_10_and_Base_2 | https://proofwiki.org/wiki/Palindromic_Primes_in_Base_10_and_Base_2 | [
"Palindromic Numbers",
"Prime Numbers",
"2",
"10"
] | [
"Definition:Prime Number",
"Definition:Palindromic Number",
"Definition:Decimal Notation",
"Definition:Binary Notation"
] | [
"Definition:Digit",
"Definition:Binary Notation"
] |
proofwiki-13277 | Ideals form Complete Lattice | Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.
Let $\II = \struct {\map {\operatorname{Ids} } L, \subseteq}$ be an inclusion ordered set,
where $\map {\operatorname{Ids} } L$ denotes the set of all ideals in $L$.
Then $\II$ is complete lattice. | Let $X \subseteq \map {\operatorname{Ids} } L$.
By Intersection of Semilattice Ideals is Ideal/Set of Sets:
:$\ds \bigcap X$ is an ideal.
By Intersection is Largest Subset/General Result:
:$\ds \forall A \in \map {\operatorname{Ids} } L: \paren {\forall I \in X: A \subseteq I} \iff A \subseteq \bigcap X$
Thus by defini... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Bounded Below Set|bounded below]] [[Definition:Join Semilattice|join semilattice]].
Let $\II = \struct {\map {\operatorname{Ids} } L, \subseteq}$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]],
where $\map {\operatorname{Ids} } L$ denotes the [[... | Let $X \subseteq \map {\operatorname{Ids} } L$.
By [[Intersection of Semilattice Ideals is Ideal/Set of Sets]]:
:$\ds \bigcap X$ is an [[Definition:Ideal (Order Theory)|ideal]].
By [[Intersection is Largest Subset/General Result]]:
:$\ds \forall A \in \map {\operatorname{Ids} } L: \paren {\forall I \in X: A \subseteq... | Ideals form Complete Lattice | https://proofwiki.org/wiki/Ideals_form_Complete_Lattice | https://proofwiki.org/wiki/Ideals_form_Complete_Lattice | [
"Complete Lattices"
] | [
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Inclusion Ordered Set",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Definition:Complete Lattice"
] | [
"Intersection of Semilattice Ideals is Ideal/Set of Sets",
"Definition:Ideal (Order Theory)",
"Intersection is Largest Subset/General Result",
"Definition:Infimum of Set",
"Lattice is Complete iff it Admits All Suprema",
"Definition:Complete Lattice"
] |
proofwiki-13278 | Filters form Complete Lattice | Let $L = \left({S, \wedge, \preceq}\right)$ be a bounded above meet semilattice.
Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an inclusion ordered set,
where $\mathit{Filt}\left({L}\right)$ denotes set of all filters on $L$.
Then $F$ is a complete lattice. | Define $L^{-1} := \left({S, \succeq}\right)$ be a dual ordered set of $L$.
By Dual Pairs (Order Theory):
:$L^{-1}$ is a bounded below meet semilattice.
By Filters equal Ideals in Dual Ordered Set:
:$\mathit{Filt}\left({L}\right) = \mathit{Ids}\left({L^{-1} }\right)$
where $\mathit{Ids}\left({L^{-1} }\right)$ denotes th... | Let $L = \left({S, \wedge, \preceq}\right)$ be a [[Definition:Bounded Above Set|bounded above]] [[Definition:Meet Semilattice|meet semilattice]].
Let $F = \left({\mathit{Filt}\left({L}\right), \subseteq}\right)$ be an [[Definition:Inclusion Ordered Set|inclusion ordered set]],
where $\mathit{Filt}\left({L}\right)$ de... | Define $L^{-1} := \left({S, \succeq}\right)$ be a [[Definition:Dual Ordered Set|dual ordered set]] of $L$.
By [[Dual Pairs (Order Theory)]]:
:$L^{-1}$ is a [[Definition:Bounded Below Set|bounded below]] [[Definition:Meet Semilattice|meet semilattice]].
By [[Filters equal Ideals in Dual Ordered Set]]:
:$\mathit{Filt}\... | Filters form Complete Lattice | https://proofwiki.org/wiki/Filters_form_Complete_Lattice | https://proofwiki.org/wiki/Filters_form_Complete_Lattice | [
"Complete Lattices"
] | [
"Definition:Bounded Above Set",
"Definition:Meet Semilattice",
"Definition:Inclusion Ordered Set",
"Definition:Set of Sets",
"Definition:Filter",
"Definition:Complete Lattice"
] | [
"Definition:Dual Ordering/Dual Ordered Set",
"Dual Pairs (Order Theory)",
"Definition:Bounded Below Set",
"Definition:Meet Semilattice",
"Filters equal Ideals in Dual Ordered Set",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)",
"Ideals form Complete Lattice",
"Definition:Complete Latti... |
proofwiki-13279 | 319 is not Expressible as Sum of Fewer than 19 Fourth Powers | :$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$
or:
:$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$ | First note that $5^4 = 625 > 319$.
Then note that $2 \times 4^4 = 512 > 319$.
Hence any expression of $319$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$.
Now we have:
{{begin-eqn}}
{{eqn |... | :$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$
or:
:$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$ | First note that $5^4 = 625 > 319$.
Then note that $2 \times 4^4 = 512 > 319$.
Hence any expression of $319$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than ... | 319 is not Expressible as Sum of Fewer than 19 Fourth Powers | https://proofwiki.org/wiki/319_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | https://proofwiki.org/wiki/319_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | [
"Fourth Powers",
"Hilbert-Waring Theorem",
"319"
] | [] | [
"Definition:Fourth Power",
"Definition:Fourth Power",
"Definition:Fourth Power"
] |
proofwiki-13280 | Existence of n such that M - 2^n or M + 2^n has no Prime factors less than 331 | Let $M \in \Z$ be an integer.
Then there exists a positive integer $n \in \Z_{\ge 0}$ such that either $M - 2^n$ or $M + 2^n$ has no prime factors less than $331$. | {{ProofWanted|Arises during the course of Not Every Number is the Sum or Difference of Two Prime Powers and may be either a lemma or corollary -- hence may require a rename. Work is needed to analyse that result.}} | Let $M \in \Z$ be an [[Definition:Integer|integer]].
Then there exists a [[Definition:Positive Integer|positive integer]] $n \in \Z_{\ge 0}$ such that either $M - 2^n$ or $M + 2^n$ has no [[Definition:Prime Factor|prime factors]] less than $331$. | {{ProofWanted|Arises during the course of [[Not Every Number is the Sum or Difference of Two Prime Powers]] and may be either a lemma or corollary -- hence may require a rename. Work is needed to analyse that result.}} | Existence of n such that M - 2^n or M + 2^n has no Prime factors less than 331 | https://proofwiki.org/wiki/Existence_of_n_such_that_M_-_2^n_or_M_+_2^n_has_no_Prime_factors_less_than_331 | https://proofwiki.org/wiki/Existence_of_n_such_that_M_-_2^n_or_M_+_2^n_has_no_Prime_factors_less_than_331 | [
"Not Every Number is the Sum or Difference of Two Prime Powers",
"331"
] | [
"Definition:Integer",
"Definition:Positive/Integer",
"Definition:Prime Factor"
] | [
"Not Every Number is the Sum or Difference of Two Prime Powers"
] |
proofwiki-13281 | Completely Irreducible implies Infimum differs from Element | Let $\struct {S, \preceq}$ be an ordered set.
Let $p \in S$ such that
:$p$ is completely irreducible.
Then $\map \inf {p^\succeq \setminus \set p} \ne p$
where $p^\succeq$ donotes the upper closure of $p$. | By definition of completely irreducible:
:$p^\succeq \setminus \set p$ admits a minimum.
Then:
:$p^\succeq \setminus \set p$ admits a infimum and $\map \inf {p^\succeq \setminus \set p} \in p^\succeq \setminus \set p$
By definition of difference:
:$\map \inf {p^\succeq \setminus \set p} \notin \set p$
Thus by definitio... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $p \in S$ such that
:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
Then $\map \inf {p^\succeq \setminus \set p} \ne p$
where $p^\succeq$ donotes the [[Definition:Upper Closure of Element|upper closure of]] $p$. | By definition of [[Definition:Completely Irreducible|completely irreducible]]:
:$p^\succeq \setminus \set p$ admits a [[Definition:Minimum Element|minimum]].
Then:
:$p^\succeq \setminus \set p$ admits a [[Definition:Infimum of Set|infimum]] and $\map \inf {p^\succeq \setminus \set p} \in p^\succeq \setminus \set p$
B... | Completely Irreducible implies Infimum differs from Element | https://proofwiki.org/wiki/Completely_Irreducible_implies_Infimum_differs_from_Element | https://proofwiki.org/wiki/Completely_Irreducible_implies_Infimum_differs_from_Element | [
"Meet Irreducible Elements"
] | [
"Definition:Ordered Set",
"Definition:Completely Irreducible",
"Definition:Upper Closure/Element"
] | [
"Definition:Completely Irreducible",
"Definition:Smallest Element",
"Definition:Infimum of Set",
"Definition:Set Difference",
"Definition:Singleton"
] |
proofwiki-13282 | Not Every Number is the Sum or Difference of Two Prime Powers | Not every positive integer can be expressed in the form $p^m \pm q^n$ where $p, q$ are prime and $m, n$ are positive integers. | What is to be demonstrated is that there exist odd integers which cannot be expressed as $2^m \pm q^n$.
{{ProofWanted|I have the paper downloaded and I am studying it.}} | Not every [[Definition:Positive Integer|positive integer]] can be expressed in the form $p^m \pm q^n$ where $p, q$ are [[Definition:Prime Number|prime]] and $m, n$ are [[Definition:Positive Integer|positive integers]]. | What is to be demonstrated is that there exist [[Definition:Odd Integer|odd integers]] which cannot be expressed as $2^m \pm q^n$.
{{ProofWanted|I have the paper downloaded and I am studying it.}} | Not Every Number is the Sum or Difference of Two Prime Powers | https://proofwiki.org/wiki/Not_Every_Number_is_the_Sum_or_Difference_of_Two_Prime_Powers | https://proofwiki.org/wiki/Not_Every_Number_is_the_Sum_or_Difference_of_Two_Prime_Powers | [
"Not Every Number is the Sum or Difference of Two Prime Powers"
] | [
"Definition:Positive/Integer",
"Definition:Prime Number",
"Definition:Positive/Integer"
] | [
"Definition:Odd Integer"
] |
proofwiki-13283 | Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $p \in S$.
Then $p$ is completely irreducible {{iff}}
:$\exists q \in S: p \prec q \land \paren {\forall s \in S: p \prec s \implies q \preceq s} \land p^\succeq = \set p \cup q^\succeq$
where $p^\succeq$ denotes the upper closure of $p$. | === Sufficient Condition ===
Assume that:
:$p$ is completely irreducible.
By definition of completely irreducible:
:$p^\succeq \setminus \set p$ admits a minimum.
By definitions of minimum and infimum:
:$p^\succeq \setminus \set p$ admits an infimum.
Define $q = \map \inf {p^\succeq \setminus \set p}$
By definition of ... | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $p \in S$.
Then $p$ is [[Definition:Completely Irreducible|completely irreducible]] {{iff}}
:$\exists q \in S: p \prec q \land \paren {\forall s \in S: p \prec s \implies q \preceq s} \land p^\succeq = \set p \cup q^\succeq$
where $p^\s... | === Sufficient Condition ===
Assume that:
:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
By definition of [[Definition:Completely Irreducible|completely irreducible]]:
:$p^\succeq \setminus \set p$ admits a [[Definition:Minimum Element|minimum]].
By definitions of [[Definition:Minimum Element|... | Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element | https://proofwiki.org/wiki/Completely_Irreducible_Element_iff_Exists_Element_that_Strictly_Succeeds_First_Element | https://proofwiki.org/wiki/Completely_Irreducible_Element_iff_Exists_Element_that_Strictly_Succeeds_First_Element | [
"Meet Irreducible Elements"
] | [
"Definition:Ordered Set",
"Definition:Completely Irreducible",
"Definition:Upper Closure/Element"
] | [
"Definition:Completely Irreducible",
"Definition:Completely Irreducible",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Infimum of Set",
"Definition:Infimum of Set",
"Definition:Smallest Element",
"Definition:Set Difference",
"Definition:Upper Closure/Element",
"Complet... |
proofwiki-13284 | Poulet Number/Examples/341 | The smallest Poulet number is $341$:
:$2^{341} \equiv 2 \pmod {341}$
despite the fact that $341$ is not prime:
:$341 = 11 \times 31$ | We have that:
{{begin-eqn}}
{{eqn | l = 2^{341}
| r = 4 \, 479 \, 489 \, 484 \, 355 \, 608 \, 421 \, 114 \, 884 \, 561 \, 136 \, 888 \, 556 \, 243 \, 290 \, 994 \, 469 \, 299 \, 069 \, 799 \, 978 \, 201 \, 927 \, 583 \, 742 \, 360 \, 321 \, 890 \, 761 \, 754 \, 986 \, 543 \, 214 \, 231 \, 552
| c =
}}
{{eq... | The smallest [[Definition:Poulet Number|Poulet number]] is $341$:
:$2^{341} \equiv 2 \pmod {341}$
despite the fact that $341$ is not [[Definition:Prime Number|prime]]:
:$341 = 11 \times 31$ | We have that:
{{begin-eqn}}
{{eqn | l = 2^{341}
| r = 4 \, 479 \, 489 \, 484 \, 355 \, 608 \, 421 \, 114 \, 884 \, 561 \, 136 \, 888 \, 556 \, 243 \, 290 \, 994 \, 469 \, 299 \, 069 \, 799 \, 978 \, 201 \, 927 \, 583 \, 742 \, 360 \, 321 \, 890 \, 761 \, 754 \, 986 \, 543 \, 214 \, 231 \, 552
| c =
}}
{{e... | Poulet Number/Examples/341 | https://proofwiki.org/wiki/Poulet_Number/Examples/341 | https://proofwiki.org/wiki/Poulet_Number/Examples/341 | [
"Poulet Numbers",
"341"
] | [
"Definition:Poulet Number",
"Definition:Prime Number"
] | [] |
proofwiki-13285 | Sums of both 2 and 3 Consecutive Squares | The following are the smallest positive integers that are the sum of both $2$ and $3$ consecutive non-zero square numbers:
:$365, 35 \, 645, 3 \, 492 \, 725, 342 \, 251 \, 285, 33 \, 537 \, 133 \, 085, 3 \, 286 \, 296 \, 790 \, 925, \ldots$
{{OEIS|A007667}} | {{begin-eqn}}
{{eqn | l = 365
| r = 10^2 + 11^2 + 12^2
| c =
}}
{{eqn | r = 13^2 + 14^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 35 \, 645
| r = 108^2 + 109^2 + 110^2
| c =
}}
{{eqn | r = 133^2 + 134^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 3 \, 492 \, 725
| r... | The following are the smallest [[Definition:Positive Integer|positive integers]] that are the [[Definition:Integer Addition|sum]] of both $2$ and $3$ consecutive non-zero [[Definition:Square Number|square numbers]]:
:$365, 35 \, 645, 3 \, 492 \, 725, 342 \, 251 \, 285, 33 \, 537 \, 133 \, 085, 3 \, 286 \, 296 \, 790 \,... | {{begin-eqn}}
{{eqn | l = 365
| r = 10^2 + 11^2 + 12^2
| c =
}}
{{eqn | r = 13^2 + 14^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 35 \, 645
| r = 108^2 + 109^2 + 110^2
| c =
}}
{{eqn | r = 133^2 + 134^2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 3 \, 492 \, 725
... | Sums of both 2 and 3 Consecutive Squares | https://proofwiki.org/wiki/Sums_of_both_2_and_3_Consecutive_Squares | https://proofwiki.org/wiki/Sums_of_both_2_and_3_Consecutive_Squares | [
"Sums of Squares"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number"
] | [] |
proofwiki-13286 | Sturm-Liouville Problem/Unit Weight Function | Let $P, Q: \R \to \R$ be real mappings such that $P$ is smooth and positive, while $Q$ is continuous:
:$\map P x \in C^\infty$
:$\map P x > 0$
:$\map Q x \in C^0$
Let the Sturm-Liouville equation, with $\map w x = 1$, be of the form:
:$-\paren {P y'}' + Q y = \lambda y$
where $\lambda \in \R$.
Let it satisfy the follo... | === Outline ===
Firstly, an equivalence between the Sturm-Liouville equation and minimisation of functional $\ds J \sqbrk y = \int_a^b \paren {P y'^2 + Q y^2} \rd x$ problems is established.
Then, the lower bound of the functional $J$ is found, thus allowing $J$ to have a finite minimisation.
Afterwards, a trial minim... | Let $P, Q: \R \to \R$ be [[Definition:Real Function|real mappings]] such that $P$ is [[Definition:Smooth Real Function|smooth]] and [[Definition:Positive Real Function|positive]], while $Q$ is [[Definition:Continuous Real Function|continuous]]:
:$\map P x \in C^\infty$
:$\map P x > 0$
:$\map Q x \in C^0$
Let the [... | === Outline ===
Firstly, an equivalence between the [[Definition:Sturm-Liouville Equation|Sturm-Liouville equation]] and [[Definition:Minimum Value of Functional|minimisation]] of [[Definition:Real Functional|functional]] $\ds J \sqbrk y = \int_a^b \paren {P y'^2 + Q y^2} \rd x$ problems is established.
Then, the [[D... | Sturm-Liouville Problem/Unit Weight Function | https://proofwiki.org/wiki/Sturm-Liouville_Problem/Unit_Weight_Function | https://proofwiki.org/wiki/Sturm-Liouville_Problem/Unit_Weight_Function | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Smooth Real Function",
"Definition:Positive Real Function",
"Definition:Continuous Real Function",
"Definition:Sturm-Liouville Equation",
"Definition:Boundary Condition",
"Definition:Differential Equation/Solution",
"Definition:Sturm-Liouville Equation",
"Defi... | [
"Definition:Sturm-Liouville Equation",
"Definition:Minimum Value of Functional",
"Definition:Functional/Real",
"Definition:Lower Bound",
"Definition:Functional/Real",
"Definition:Minimum Value of Functional",
"Definition:Sequence/Minimizing/Functional",
"Definition:Function",
"Definition:Coefficient... |
proofwiki-13287 | Completely Irreducible implies Meet Irreducible | Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $p \in S$.
Then if $p$ is completely irreducible, then $p$ is meet irreducible. | Assume that
:$p$ is completely irreducible.
By Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element
:$\exists q \in S: p \prec q \land \left({\forall s \in S: p \prec s \implies q \preceq s}\right) \land p^\succeq = \left\{ {p}\right\} \cup q^\succeq$
where $p^\succeq$ denotes the uppe... | Let $L = \struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $p \in S$.
Then if $p$ is [[Definition:Completely Irreducible|completely irreducible]], then $p$ is [[Definition:Meet Irreducible Element|meet irreducible]]. | Assume that
:$p$ is [[Definition:Completely Irreducible|completely irreducible]].
By [[Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element]]
:$\exists q \in S: p \prec q \land \left({\forall s \in S: p \prec s \implies q \preceq s}\right) \land p^\succeq = \left\{ {p}\right\} \cup q^... | Completely Irreducible implies Meet Irreducible | https://proofwiki.org/wiki/Completely_Irreducible_implies_Meet_Irreducible | https://proofwiki.org/wiki/Completely_Irreducible_implies_Meet_Irreducible | [
"Meet Irreducible Elements"
] | [
"Definition:Meet Semilattice",
"Definition:Completely Irreducible",
"Definition:Meet Irreducible Element"
] | [
"Definition:Completely Irreducible",
"Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element",
"Definition:Upper Closure/Element",
"Meet Precedes Operands",
"Definition:Strictly Precede",
"Definition:Strictly Precede",
"Meet Precedes Operands",
"Definition:Strictly Prec... |
proofwiki-13288 | Discrepancy between Julian Year and Tropical Year | The Julian year and the tropical year differ such that the Julian calendar becomes $1$ day further out approximately every $128$ years. | By definition, the length $Y_T$ of the tropical year is defined as $\approx 365 \cdotp 24219 \, 878$ days
By definition of the Julian year:
:$Y_J = \begin{cases} 366 \, \text {days} & : 4 \divides y \\
365 \, \text {days} & : 4 \nmid y \end{cases}$
where:
:$Y_J$ denotes the length of the Julian year in days
:$y$ denote... | The [[Definition:Julian Year|Julian year]] and the [[Definition:Tropical Year|tropical year]] differ such that the [[Definition:Julian Calendar|Julian calendar]] becomes $1$ [[Definition:Day|day]] further out approximately every $128$ [[Definition:Year|years]]. | By definition, the [[Definition:Length of Time|length]] $Y_T$ of the [[Definition:Tropical Year|tropical year]] is defined as $\approx 365 \cdotp 24219 \, 878$ [[Definition:Day|days]]
By definition of the [[Definition:Julian Year|Julian year]]:
:$Y_J = \begin{cases} 366 \, \text {days} & : 4 \divides y \\
365 \, \text... | Discrepancy between Julian Year and Tropical Year | https://proofwiki.org/wiki/Discrepancy_between_Julian_Year_and_Tropical_Year | https://proofwiki.org/wiki/Discrepancy_between_Julian_Year_and_Tropical_Year | [
"Calendars"
] | [
"Definition:Calendar/Julian/Year",
"Definition:Tropical Year",
"Definition:Calendar/Julian",
"Definition:Time/Unit/Day",
"Definition:Time/Unit/Year"
] | [
"Definition:Time/Length",
"Definition:Tropical Year",
"Definition:Time/Unit/Day",
"Definition:Calendar/Julian/Year",
"Definition:Time/Length",
"Definition:Calendar/Julian/Year",
"Definition:Time/Unit/Day",
"Definition:Time/Unit/Year",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Al... |
proofwiki-13289 | Discrepancy between Gregorian Year and Tropical Year | The Gregorian year and the tropical year differ such that the Gregorian calendar becomes $1$ day further out approximately every $3319$ years. | By definition, the length $Y_T$ of the tropical year is defined as $\approx 365 \cdotp 24219 \, 878$ days.
By definition of the Gregorian year:
:$Y_G = \begin{cases} 366 \, \text {days} & : 400 \divides y \\
365 \, \text {days} & : 400 \nmid y \text { and } 100 \divides y\\
366 \, \text {days} & : 100 \nmid y \text {... | The [[Definition:Gregorian Year|Gregorian year]] and the [[Definition:Tropical Year|tropical year]] differ such that the [[Definition:Gregorian Calendar|Gregorian calendar]] becomes $1$ [[Definition:Day|day]] further out approximately every $3319$ [[Definition:Year|years]]. | By definition, the [[Definition:Length of Time|length]] $Y_T$ of the [[Definition:Tropical Year|tropical year]] is defined as $\approx 365 \cdotp 24219 \, 878$ [[Definition:Day|days]].
By definition of the [[Definition:Gregorian Year|Gregorian year]]:
:$Y_G = \begin{cases} 366 \, \text {days} & : 400 \divides y \\
365... | Discrepancy between Gregorian Year and Tropical Year | https://proofwiki.org/wiki/Discrepancy_between_Gregorian_Year_and_Tropical_Year | https://proofwiki.org/wiki/Discrepancy_between_Gregorian_Year_and_Tropical_Year | [
"Calendars"
] | [
"Definition:Calendar/Gregorian/Year",
"Definition:Tropical Year",
"Definition:Calendar/Gregorian",
"Definition:Time/Unit/Day",
"Definition:Time/Unit/Year"
] | [
"Definition:Time/Length",
"Definition:Tropical Year",
"Definition:Time/Unit/Day",
"Definition:Calendar/Gregorian/Year",
"Definition:Time/Length",
"Definition:Calendar/Gregorian/Year",
"Definition:Time/Unit/Day",
"Definition:Time/Unit/Year",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divis... |
proofwiki-13290 | Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset | Let $L = \struct {S, \preceq}$ be an ordered set.
Let $x \in S$ such that
:$x$ is completely irreducible.
Let $X \subseteq S$ such that
:$X$ admits an infimum and $x = \inf X$
Then $x \in X$ | {{AimForCont}}
:$x \notin X$
By Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element
:$\exists q \in S: x \prec q \land \left({\forall s \in S: x \prec s \implies q \preceq s}\right) \land x^\succeq = \left\{ {x}\right\} \cup q^\succeq$
where $x^\succeq$ denotes the upper closure of $x... | Let $L = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x \in S$ such that
:$x$ is [[Definition:Completely Irreducible|completely irreducible]].
Let $X \subseteq S$ such that
:$X$ admits an [[Definition:Infimum of Set|infimum]] and $x = \inf X$
Then $x \in X$ | {{AimForCont}}
:$x \notin X$
By [[Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element]]
:$\exists q \in S: x \prec q \land \left({\forall s \in S: x \prec s \implies q \preceq s}\right) \land x^\succeq = \left\{ {x}\right\} \cup q^\succeq$
where $x^\succeq$ denotes the [[Definition:U... | Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset | https://proofwiki.org/wiki/Completely_Irreducible_and_Subset_Admits_Infimum_Equals_Element_implies_Element_Belongs_to_Subset | https://proofwiki.org/wiki/Completely_Irreducible_and_Subset_Admits_Infimum_Equals_Element_implies_Element_Belongs_to_Subset | [
"Meet Irreducible Elements"
] | [
"Definition:Ordered Set",
"Definition:Completely Irreducible",
"Definition:Infimum of Set"
] | [
"Completely Irreducible Element iff Exists Element that Strictly Succeeds First Element",
"Definition:Upper Closure/Element",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Strictly Precede",
"Definition:Upper Closure/Element",
"Infimum of Upper Closure of Element",
"Infimum... |
proofwiki-13291 | Order Generating Subset Includes Completely Irreducible Elements | Let $\struct {S, \preceq}$ be an ordered set.
Let $X \subseteq S$ be an order generating subset of $S$.
Let $x \in S$ be a completely irreducible element of $S$.
Then $x \in X$. | By definition of order generating:
:$x^\succeq \cap X$ admits an infimum and $x = \map \inf {x^\succeq \cap X}$
By Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset:
:$x \in x^\succeq \cap X$
Thus by definition of intersection:
:$x \in X$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X \subseteq S$ be an [[Definition:Order Generating Subset|order generating subset]] of $S$.
Let $x \in S$ be a [[Definition:Completely Irreducible|completely irreducible]] [[Definition:Element|element]] of $S$.
Then $x \in X$. | By definition of [[Definition:Order Generating Subset|order generating]]:
:$x^\succeq \cap X$ admits an [[Definition:Infimum of Set|infimum]] and $x = \map \inf {x^\succeq \cap X}$
By [[Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset]]:
:$x \in x^\succeq \cap X$
Thus ... | Order Generating Subset Includes Completely Irreducible Elements | https://proofwiki.org/wiki/Order_Generating_Subset_Includes_Completely_Irreducible_Elements | https://proofwiki.org/wiki/Order_Generating_Subset_Includes_Completely_Irreducible_Elements | [
"Meet Irreducible Elements",
"Order Generating Subsets"
] | [
"Definition:Ordered Set",
"Definition:Order Generating Subset",
"Definition:Completely Irreducible",
"Definition:Element"
] | [
"Definition:Order Generating Subset",
"Definition:Infimum of Set",
"Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset",
"Definition:Set Intersection"
] |
proofwiki-13292 | 399 is not Expressible as Sum of Fewer than 19 Fourth Powers | :$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$
or:
:$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$ | First note that $5^4 = 625 > 399$.
Then note that $2 \times 4^4 = 512 > 399$.
Hence any expression of $399$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$ does.
Now we have:
{{begin-eqn}}
{{... | :$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$
or:
:$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$ | First note that $5^4 = 625 > 399$.
Then note that $2 \times 4^4 = 512 > 399$.
Hence any expression of $399$ as [[Definition:Fourth Power|fourth powers]] uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer [[Definition:Fourth Power|fourth powers]] than ... | 399 is not Expressible as Sum of Fewer than 19 Fourth Powers | https://proofwiki.org/wiki/399_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | https://proofwiki.org/wiki/399_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | [
"Fourth Powers",
"Hilbert-Waring Theorem",
"399"
] | [] | [
"Definition:Fourth Power",
"Definition:Fourth Power",
"Definition:Fourth Power"
] |
proofwiki-13293 | Maximal implies Difference equals Intersection | Let $\left({S, \preceq}\right)$ be an ordered set.
Let $x, y \in S$ such that
:$x$ is maximal in $S \setminus y^\succeq$
Then $x^\succeq \setminus \left\{ {x}\right\} = x^\succeq \cap y^\succeq$ | By definition of maximal:
:$x \in S \setminus y^\succeq$
By definition of difference:
:$x \notin y^\succeq$
By definition of upper closure of element:
:$y \npreceq x$
We will prove that
:$x^\succeq \setminus \left\{ {x}\right\} \subseteq x^\succeq \cap y^\succeq$
Let $a \in x^\succeq \setminus \left\{ {x}\right\}$
By d... | Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$ such that
:$x$ is [[Definition:Maximal|maximal]] in $S \setminus y^\succeq$
Then $x^\succeq \setminus \left\{ {x}\right\} = x^\succeq \cap y^\succeq$ | By definition of [[Definition:Maximal|maximal]]:
:$x \in S \setminus y^\succeq$
By definition of [[Definition:Set Difference|difference]]:
:$x \notin y^\succeq$
By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
:$y \npreceq x$
We will prove that
:$x^\succeq \setminus \left\{ {x}\righ... | Maximal implies Difference equals Intersection | https://proofwiki.org/wiki/Maximal_implies_Difference_equals_Intersection | https://proofwiki.org/wiki/Maximal_implies_Difference_equals_Intersection | [
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Maximal"
] | [
"Definition:Maximal",
"Definition:Set Difference",
"Definition:Upper Closure/Element",
"Definition:Set Difference",
"Definition:Upper Closure/Element",
"Definition:Singleton",
"Definition:Strictly Precede",
"Definition:Maximal",
"Definition:Set Difference",
"Definition:Set Intersection",
"Defini... |
proofwiki-13294 | Square of n such that 2n-1 is Composite is not Sum of Square and Prime | Let $n^2$ be a square such that $2 n - 1$ is composite.
Then $n^2$ cannot be expressed as the sum of a square and a prime. | The case where $n = 1$ is trivial, as there are no prime numbers less than $1$.
Let $n, m \in \Z$ be integers such that $n > 1$.
Let $n^2 = m^2 + p$ where $p$ is prime.
Then:
{{begin-eqn}}
{{eqn | l = p
| r = n^2 - m^2
| c =
}}
{{eqn | r = \paren {n + m} \paren {n - m}
| c = Difference of Two Squares... | Let $n^2$ be a [[Definition:Square Number|square]] such that $2 n - 1$ is [[Definition:Composite Number|composite]].
Then $n^2$ cannot be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Square Number|square]] and a [[Definition:Prime Number|prime]]. | The case where $n = 1$ is trivial, as there are no [[Definition:Prime Number|prime numbers]] less than $1$.
Let $n, m \in \Z$ be [[Definition:Integer|integers]] such that $n > 1$.
Let $n^2 = m^2 + p$ where $p$ is [[Definition:Prime Number|prime]].
Then:
{{begin-eqn}}
{{eqn | l = p
| r = n^2 - m^2
| c =... | Square of n such that 2n-1 is Composite is not Sum of Square and Prime | https://proofwiki.org/wiki/Square_of_n_such_that_2n-1_is_Composite_is_not_Sum_of_Square_and_Prime | https://proofwiki.org/wiki/Square_of_n_such_that_2n-1_is_Composite_is_not_Sum_of_Square_and_Prime | [
"Numbers not Sum of Square and Prime"
] | [
"Definition:Square Number",
"Definition:Composite Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Integer",
"Definition:Prime Number",
"Difference of Two Squares",
"Definition:Composite Number",
"Definition:Prime Number",
"Category:Numbers not Sum of Square and Prime"
] |
proofwiki-13295 | 40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7 | Consider the heptagonal numbers:
:$\ds H_n = \sum_{k \mathop = 1}^n \paren {5 k - 4}$
Let $S_n$ be the sequence defined as:
:$\forall n \in \Z_{>1}: S_n = 40 \times H_n + 9$
Then $S_n$ consists of the squares of all the positive integers which end in a $7$:
:$49, 289, 729, 1369, 2209, 3249, 4489, 5929, 7569, \ldots$
th... | For $n \in \Z_{>1}$, we have:
{{begin-eqn}}
{{eqn | l = 10 \times H_n + 9
| r = 40 \dfrac {n \paren {5 n - 3} } 2 + 9
| c = Closed Form for Heptagonal Numbers
}}
{{eqn | r = 20 n \paren {5 n - 3} + 9
| c =
}}
{{eqn | r = 100 n^2 - 60 n + 9
| c =
}}
{{eqn | r = \paren {10 n - 3}^2
| c =
... | Consider the [[Definition:Heptagonal Number|heptagonal numbers]]:
:$\ds H_n = \sum_{k \mathop = 1}^n \paren {5 k - 4}$
Let $S_n$ be the [[Definition:Integer Sequence|sequence]] defined as:
:$\forall n \in \Z_{>1}: S_n = 40 \times H_n + 9$
Then $S_n$ consists of the [[Definition:Square (Algebra)|squares]] of all th... | For $n \in \Z_{>1}$, we have:
{{begin-eqn}}
{{eqn | l = 10 \times H_n + 9
| r = 40 \dfrac {n \paren {5 n - 3} } 2 + 9
| c = [[Closed Form for Heptagonal Numbers]]
}}
{{eqn | r = 20 n \paren {5 n - 3} + 9
| c =
}}
{{eqn | r = 100 n^2 - 60 n + 9
| c =
}}
{{eqn | r = \paren {10 n - 3}^2
| ... | 40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7 | https://proofwiki.org/wiki/40_times_Heptagonal_Numbers_plus_9_gives_Squares_of_Numbers_ending_in_7 | https://proofwiki.org/wiki/40_times_Heptagonal_Numbers_plus_9_gives_Squares_of_Numbers_ending_in_7 | [
"Heptagonal Numbers",
"Square Numbers"
] | [
"Definition:Heptagonal Number",
"Definition:Integer Sequence",
"Definition:Square/Function",
"Definition:Positive/Integer"
] | [
"Closed Form for Heptagonal Numbers",
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Category:Heptagonal Numbers",
"Category:Square Numbers"
] |
proofwiki-13296 | Maximal implies Completely Irreducible | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $p \in S$ such that
:$\exists k \in S: p$ is maximal in $S \setminus k^\succeq$
Then $p$ is completely irreducible. | By Join Succeeds Operands:
:$k \preceq p \vee k$ and $p \preceq p \vee k$
By definition of upper closure of element:
:$p \vee k \in k^\succeq$ and $p \vee k \in p^\succeq$
By definition of intersection:
:$p \vee k \in p^\succeq \cap k^\succeq$
By Maximal implies Difference equals Intersection:
:$p^\succeq \setminus \se... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $p \in S$ such that
:$\exists k \in S: p$ is [[Definition:Maximal Element|maximal]] in $S \setminus k^\succeq$
Then $p$ is [[Definition:Completely Irreducible|completely irreducible]]. | By [[Join Succeeds Operands]]:
:$k \preceq p \vee k$ and $p \preceq p \vee k$
By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
:$p \vee k \in k^\succeq$ and $p \vee k \in p^\succeq$
By definition of [[Definition:Set Intersection|intersection]]:
:$p \vee k \in p^\succeq \cap k^\succeq... | Maximal implies Completely Irreducible | https://proofwiki.org/wiki/Maximal_implies_Completely_Irreducible | https://proofwiki.org/wiki/Maximal_implies_Completely_Irreducible | [
"Meet Irreducible Elements"
] | [
"Definition:Complete Lattice",
"Definition:Maximal/Element",
"Definition:Completely Irreducible"
] | [
"Join Succeeds Operands",
"Definition:Upper Closure/Element",
"Definition:Set Intersection",
"Maximal implies Difference equals Intersection",
"Infimum of Intersection of Upper Closures equals Join Operands",
"Definition:Smallest Element",
"Definition:Completely Irreducible"
] |
proofwiki-13297 | Infimum of Intersection of Upper Closures equals Join Operands | Let $L = \struct {S, \vee, \preceq}$ be a join semilattice.
Let $x, y \in S$.
Then $\map \inf {x^\succeq \cap y^\succeq} = x \vee y$ | By Intersection of Upper Closures is Upper Closure of Join Operands:
:$x^\succeq \cap y^\succeq = \paren {x \vee y}^\succeq$
Thus by Infimum of Upper Closure of Element:
:$\map \inf {x^\succeq \cap y^\succeq} = x \vee y$
{{qed}} | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $x, y \in S$.
Then $\map \inf {x^\succeq \cap y^\succeq} = x \vee y$ | By [[Intersection of Upper Closures is Upper Closure of Join Operands]]:
:$x^\succeq \cap y^\succeq = \paren {x \vee y}^\succeq$
Thus by [[Infimum of Upper Closure of Element]]:
:$\map \inf {x^\succeq \cap y^\succeq} = x \vee y$
{{qed}} | Infimum of Intersection of Upper Closures equals Join Operands | https://proofwiki.org/wiki/Infimum_of_Intersection_of_Upper_Closures_equals_Join_Operands | https://proofwiki.org/wiki/Infimum_of_Intersection_of_Upper_Closures_equals_Join_Operands | [
"Upper Closures"
] | [
"Definition:Join Semilattice"
] | [
"Intersection of Upper Closures is Upper Closure of Join Operands",
"Infimum of Upper Closure of Element"
] |
proofwiki-13298 | Sturm-Liouville Problem/Unit Weight Function/Lemma | Let $\map \alpha x: \R \to \R$ such that $\map \alpha x \in C^2 \closedint a b$.
Suppose:
:$\ds \forall \map h x \in C^2 \closedint a b: \map h a = \map h b = \map {h'} a = \map {h'} b = 0: \int_a^b \map \alpha x \, \map {h''} x \rd x = 0$
Then:
:$\forall x \in \closedint a b: \map \alpha x = c_0 + c_1 x$
where $ c_0, ... | Let $ c_0, c_1$ be defined by the conditions:
:$\ds \int_a^b \paren {\map \alpha x - c_0 - c_1 x} \rd x = 0$
:$\ds \int_a^b \rd x \int_a^x \paren {\map \alpha \xi - c_0 - c_1 \xi} \rd \xi = 0$
Suppose:
:$\ds \map h x = \int_a^x \xi \int_a^\xi \paren {\map \alpha t - c_0 - c_1 t} \rd t$
This form satisfies conditions on... | Let $\map \alpha x: \R \to \R$ such that $\map \alpha x \in C^2 \closedint a b$.
Suppose:
:$\ds \forall \map h x \in C^2 \closedint a b: \map h a = \map h b = \map {h'} a = \map {h'} b = 0: \int_a^b \map \alpha x \, \map {h''} x \rd x = 0$
Then:
:$\forall x \in \closedint a b: \map \alpha x = c_0 + c_1 x$
where $... | Let $ c_0, c_1$ be defined by the conditions:
:$\ds \int_a^b \paren {\map \alpha x - c_0 - c_1 x} \rd x = 0$
:$\ds \int_a^b \rd x \int_a^x \paren {\map \alpha \xi - c_0 - c_1 \xi} \rd \xi = 0$
Suppose:
:$\ds \map h x = \int_a^x \xi \int_a^\xi \paren {\map \alpha t - c_0 - c_1 t} \rd t$
This form satisfies conditio... | Sturm-Liouville Problem/Unit Weight Function/Lemma | https://proofwiki.org/wiki/Sturm-Liouville_Problem/Unit_Weight_Function/Lemma | https://proofwiki.org/wiki/Sturm-Liouville_Problem/Unit_Weight_Function/Lemma | [
"Calculus of Variations"
] | [] | [
"Category:Calculus of Variations"
] |
proofwiki-13299 | Intersection of Upper Closures is Upper Closure of Join Operands | Let $L = \struct {S, \vee, \preceq}$ be a join semilattice.
Let $x, y \in S$.
Then $\paren {x \vee y}^\succeq = x^\succeq \cap y^\succeq$ | We will prove that:
:$\paren {x \vee y}^\succeq \subseteq x^\succeq \cap y^\succeq$
Let $a \in \paren {x \vee y}^\succeq$
By definition of upper closure of element:
:$x \vee y \preceq a$
By Join Succeeds Operands:
:$x \preceq x \vee y$ and $y \preceq x \vee y$
By definition of transitivity:
:$x \preceq a$ and $y \prece... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $x, y \in S$.
Then $\paren {x \vee y}^\succeq = x^\succeq \cap y^\succeq$ | We will prove that:
:$\paren {x \vee y}^\succeq \subseteq x^\succeq \cap y^\succeq$
Let $a \in \paren {x \vee y}^\succeq$
By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
:$x \vee y \preceq a$
By [[Join Succeeds Operands]]:
:$x \preceq x \vee y$ and $y \preceq x \vee y$
By definiti... | Intersection of Upper Closures is Upper Closure of Join Operands | https://proofwiki.org/wiki/Intersection_of_Upper_Closures_is_Upper_Closure_of_Join_Operands | https://proofwiki.org/wiki/Intersection_of_Upper_Closures_is_Upper_Closure_of_Join_Operands | [
"Upper Closures"
] | [
"Definition:Join Semilattice"
] | [
"Definition:Upper Closure/Element",
"Join Succeeds Operands",
"Definition:Transitive",
"Definition:Upper Closure/Element",
"Definition:Set Intersection",
"Definition:Set Intersection",
"Definition:Upper Closure/Element",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Up... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.