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-13600 | Factors of Sums of Powers of 100,000 | All integers $n$ of the form:
:$n = \ds \sum_{k \mathop = 0}^m 10^{5 k}$ for $m \in \Z_{> 0}$
are composite. | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^1 10^{5 k}
| r = 100 \, 001
| c =
}}
{{eqn | r = 11 \times 9091
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^2 10^{5 k}
| r = 10 \, 000 \, 100 \, 001
| c =
}}
{{eqn | r = 3 \times 31 \times 37 \times 2 \, 906 \, 161
... | All [[Definition:Integer|integers]] $n$ of the form:
:$n = \ds \sum_{k \mathop = 0}^m 10^{5 k}$ for $m \in \Z_{> 0}$
are [[Definition:Composite Number|composite]]. | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^1 10^{5 k}
| r = 100 \, 001
| c =
}}
{{eqn | r = 11 \times 9091
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^2 10^{5 k}
| r = 10 \, 000 \, 100 \, 001
| c =
}}
{{eqn | r = 3 \times 31 \times 37 \times 2 \, 906 \, 161... | Factors of Sums of Powers of 100,000 | https://proofwiki.org/wiki/Factors_of_Sums_of_Powers_of_100,000 | https://proofwiki.org/wiki/Factors_of_Sums_of_Powers_of_100,000 | [
"Composite Numbers",
"Factors of Sums of Powers of 100,000"
] | [
"Definition:Integer",
"Definition:Composite Number"
] | [
"Definition:Repunit",
"Definition:Prime Number",
"Divisors of Repunit with Composite Index",
"Prime not Divisor implies Coprime",
"Definition:Coprime/Integers",
"Condition for Repunits to be Coprime",
"Definition:Coprime/Integers",
"Euclid's Lemma",
"Divisors of Repunit with Composite Index"
] |
proofwiki-13601 | Numbers whose Cube equals Sum of Sequence of that many Squares | The integers $m$ in the following sequence all have the property that $m^3$ is equal to the sum of $m$ consecutive squares:
:$m^3 = \ds \sum_{k \mathop = 1}^m \paren {n + k}^2$
for some $n \in \Z_{\ge 0}$:
:$0, 1, 47, 2161, 99 \, 359, 4 \, 568 \, 353, \ldots$ | We have:
{{begin-eqn}}
{{eqn | n = 1
| l = m^3
| r = \sum_{k \mathop = 1}^m \paren {n + k}^2
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^m \paren {n^2 + 2 n k + k^2}
| c =
}}
{{eqn | r = n^2 \sum_{k \mathop = 1}^m 1 + 2 n \sum_{k \mathop = 1}^m k + \sum_{k \mathop = 1}^m k^2
| c =
}}
{{e... | The [[Definition:Integer|integers]] $m$ in the following [[Definition:Integer Sequence|sequence]] all have the property that $m^3$ is equal to the [[Definition:Integer Addition|sum]] of $m$ consecutive [[Definition:Square Number|squares]]:
:$m^3 = \ds \sum_{k \mathop = 1}^m \paren {n + k}^2$
for some $n \in \Z_{\ge 0... | We have:
{{begin-eqn}}
{{eqn | n = 1
| l = m^3
| r = \sum_{k \mathop = 1}^m \paren {n + k}^2
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^m \paren {n^2 + 2 n k + k^2}
| c =
}}
{{eqn | r = n^2 \sum_{k \mathop = 1}^m 1 + 2 n \sum_{k \mathop = 1}^m k + \sum_{k \mathop = 1}^m k^2
| c =
}}
{{... | Numbers whose Cube equals Sum of Sequence of that many Squares | https://proofwiki.org/wiki/Numbers_whose_Cube_equals_Sum_of_Sequence_of_that_many_Squares | https://proofwiki.org/wiki/Numbers_whose_Cube_equals_Sum_of_Sequence_of_that_many_Squares | [
"Numbers whose Cube equals Sum of Sequence of that many Squares",
"Cube Numbers",
"Square Numbers"
] | [
"Definition:Integer",
"Definition:Integer Sequence",
"Definition:Addition/Integers",
"Definition:Square Number"
] | [
"Closed Form for Triangular Numbers",
"Sum of Sequence of Squares",
"Definition:Quadratic Equation",
"Solution to Quadratic Equation",
"Definition:Integer",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Square Root",
"Definition:Integer",
"Definition:Integer",
"Definition:Integ... |
proofwiki-13602 | Points Defined by Adjacent Pairs of Digits of Reciprocal of 7 lie on Ellipse | Consider the digits that form the recurring part of the reciprocal of $7$:
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$
Take the digits in ordered pairs, and treat them as coordinates of a Cartesian plane.
It will be found that they all lie on an ellipse:
:400px | :400px
Let the points be labelled to simplify:
:$A := \tuple {1, 4}$
:$B := \tuple {2, 8}$
:$C := \tuple {4, 2}$
:$D := \tuple {8, 5}$
:$E := \tuple {7, 1}$
:$F := \tuple {5, 7}$
Let $ABCDEF$ be considered as a hexagon.
We join the opposite points of $ABCDEF$:
:$AF: \tuple {1, 4} \to \tuple {5, 7}$
:$BC: \tuple {2, 8} ... | Consider the [[Definition:Digit|digits]] that form the [[Definition:Recurring Part|recurring part]] of the [[Definition:Reciprocal|reciprocal]] of $7$:
:$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$
Take the [[Definition:Digit|digits]] in [[Definition:Ordered Pair|ordered pairs]], and treat them as [[Definition:Coordinate... | :[[File:EllipseFromSeventhSolution.png|400px]]
Let the points be labelled to simplify:
:$A := \tuple {1, 4}$
:$B := \tuple {2, 8}$
:$C := \tuple {4, 2}$
:$D := \tuple {8, 5}$
:$E := \tuple {7, 1}$
:$F := \tuple {5, 7}$
Let $ABCDEF$ be considered as a [[Definition:Hexagon|hexagon]].
We join the opposite points of ... | Points Defined by Adjacent Pairs of Digits of Reciprocal of 7 lie on Ellipse | https://proofwiki.org/wiki/Points_Defined_by_Adjacent_Pairs_of_Digits_of_Reciprocal_of_7_lie_on_Ellipse | https://proofwiki.org/wiki/Points_Defined_by_Adjacent_Pairs_of_Digits_of_Reciprocal_of_7_lie_on_Ellipse | [
"Ellipses",
"7"
] | [
"Definition:Digit",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Reciprocal",
"Definition:Digit",
"Definition:Ordered Pair",
"Definition:Coordinate",
"Definition:Cartesian Plane",
"Definition:Ellipse",
"File:EllipseFromSeventh.png"
] | [
"File:EllipseFromSeventhSolution.png",
"Definition:Hexagon",
"Definition:Intersection (Geometry)",
"Definition:Line/Straight Line",
"Pascal's Mystic Hexagram",
"Equation of Straight Line in Plane/Two-Point Form",
"Definition:Intersection (Geometry)",
"Definition:Intersection (Geometry)",
"Definition... |
proofwiki-13603 | Continuous Function on Compact Subspace of Euclidean Space is Bounded | Let $\R^n$ be the $n$-dimensional Euclidean space.
Let $S \subseteq \R^n$ be a compact subspace of $\R^n$.
Let $f: S \to \R$ be a continuous function.
Then $f$ is bounded in $\R$. | An application of Continuous Function on Compact Space is Bounded. | Let $\R^n$ be the [[Definition:Euclidean Space|$n$-dimensional Euclidean space]].
Let $S \subseteq \R^n$ be a [[Definition:Compact (Real Analysis)|compact subspace]] of $\R^n$.
Let $f: S \to \R$ be a [[Definition:Continuous Mapping (Metric Spaces)|continuous function]].
Then $f$ is [[Definition:Bounded Mapping|boun... | An application of [[Continuous Function on Compact Space is Bounded]]. | Continuous Function on Compact Subspace of Euclidean Space is Bounded | https://proofwiki.org/wiki/Continuous_Function_on_Compact_Subspace_of_Euclidean_Space_is_Bounded | https://proofwiki.org/wiki/Continuous_Function_on_Compact_Subspace_of_Euclidean_Space_is_Bounded | [
"Analysis",
"Continuity"
] | [
"Definition:Euclidean Space",
"Definition:Compact Space/Real Analysis",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Bounded Mapping"
] | [
"Continuous Function on Compact Space is Bounded"
] |
proofwiki-13604 | Points Defined by Adjacent Pairs of Digits of Reciprocal of 13 lie on Hyperbola | Consider the digits that form the recurring part of the reciprocal of $13$:
:$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3$
Take the digits in ordered pairs, and treat them as coordinates of a Cartesian plane.
It will be found that they all lie on a hyperbola:
:600px | :600px
Let the points be labelled to simplify:
:$A := \left({0, 7}\right)$
:$B := \left({7, 6}\right)$
:$C := \left({6, 9}\right)$
:$D := \left({9, 2}\right)$
:$E := \left({2, 3}\right)$
:$F := \left({3, 0}\right)$
{{finish|Just too tedious to contemplate.}} | Consider the [[Definition:Digit|digits]] that form the [[Definition:Recurring Part|recurring part]] of the [[Definition:Reciprocal|reciprocal]] of $13$:
:$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3$
Take the [[Definition:Digit|digits]] in [[Definition:Ordered Pair|ordered pairs]], and treat them as [[Definition:Coordi... | :[[File:HyperbolaFromThirteenthSolution.png|600px]]
Let the points be labelled to simplify:
:$A := \left({0, 7}\right)$
:$B := \left({7, 6}\right)$
:$C := \left({6, 9}\right)$
:$D := \left({9, 2}\right)$
:$E := \left({2, 3}\right)$
:$F := \left({3, 0}\right)$
{{finish|Just too tedious to contemplate.}} | Points Defined by Adjacent Pairs of Digits of Reciprocal of 13 lie on Hyperbola | https://proofwiki.org/wiki/Points_Defined_by_Adjacent_Pairs_of_Digits_of_Reciprocal_of_13_lie_on_Hyperbola | https://proofwiki.org/wiki/Points_Defined_by_Adjacent_Pairs_of_Digits_of_Reciprocal_of_13_lie_on_Hyperbola | [
"Hyperbolas",
"13"
] | [
"Definition:Digit",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Reciprocal",
"Definition:Digit",
"Definition:Ordered Pair",
"Definition:Coordinate",
"Definition:Cartesian Plane",
"Definition:Hyperbola",
"File:HyperbolaFromThirteenth.png"
] | [
"File:HyperbolaFromThirteenthSolution.png"
] |
proofwiki-13605 | Number times Recurring Part of Reciprocal gives 9-Repdigit | Let a (strictly) positive integer $n$ be such that the decimal expansion of its reciprocal has a recurring part of period $d$ and no non-recurring part.
Let $m$ be the integer formed from the $d$ digits of the recurring part.
Then $m \times n$ is a $d$-digit repdigit number consisting of $9$s. | Let $x = \dfrac 1 n = \sqbrk {0. mmmm \dots}$.
Then:
:$10^d x = \sqbrk {m.mmmm \dots}$
Therefore:
{{begin-eqn}}
{{eqn | l = 10^d x - x
| r = \sqbrk {m.mmmm \dots} - \sqbrk {0. mmmm \dots}
}}
{{eqn | ll= \leadsto
| l = \frac 1 n \paren {10^d - 1}
| r = m
}}
{{eqn | ll= \leadsto
| l = m n
| ... | Let a [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$ be such that the [[Definition:Decimal Expansion|decimal expansion]] of its [[Definition:Reciprocal|reciprocal]] has a [[Definition:Recurring Part|recurring part]] of [[Definition:Period of Recurrence|period]] $d$ and no [[Definition:Non-Recu... | Let $x = \dfrac 1 n = \sqbrk {0. mmmm \dots}$.
Then:
:$10^d x = \sqbrk {m.mmmm \dots}$
Therefore:
{{begin-eqn}}
{{eqn | l = 10^d x - x
| r = \sqbrk {m.mmmm \dots} - \sqbrk {0. mmmm \dots}
}}
{{eqn | ll= \leadsto
| l = \frac 1 n \paren {10^d - 1}
| r = m
}}
{{eqn | ll= \leadsto
| l = m n
... | Number times Recurring Part of Reciprocal gives 9-Repdigit | https://proofwiki.org/wiki/Number_times_Recurring_Part_of_Reciprocal_gives_9-Repdigit | https://proofwiki.org/wiki/Number_times_Recurring_Part_of_Reciprocal_gives_9-Repdigit | [
"Number times Recurring Part of Reciprocal gives 9-Repdigit",
"Repdigit Numbers",
"Reciprocals"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Basis Expansion/Recurrence/Non-Recurring Part",
"Definition:Integer",
"Definition:Digit",... | [
"Definition:Digit",
"Definition:Repdigit Number"
] |
proofwiki-13606 | Reciprocal of 142,857 | :$\dfrac 1 {142 \, 857} = 0 \cdotp \dot 00000 \, \dot 7$ | Performing the calculation using long division:
<pre>
0.000007000007...
------------------
142857)1.000000000000...
1 000000
999999
--------
1000000
999999
-------
</pre> | :$\dfrac 1 {142 \, 857} = 0 \cdotp \dot 00000 \, \dot 7$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.000007000007...
------------------
142857)1.000000000000...
1 000000
999999
--------
1000000
999999
-------
</pre> | Reciprocal of 142,857 | https://proofwiki.org/wiki/Reciprocal_of_142,857 | https://proofwiki.org/wiki/Reciprocal_of_142,857 | [
"142,857",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division"
] |
proofwiki-13607 | Quotient of Group by Itself | Let $G$ be a group.
Let $G / G$ be the quotient group of $G$ by itself.
Then:
:$G / G \cong \set e$
That is, the quotient of a group by itself is isomorphic to the trivial group. | Let the homomorphism $\phi: G \to \set e$ be defined as:
:$\forall g \in G: \map \phi g = e$
Then:
:$\map \ker \phi = G$
and:
:$\Img \phi = \set e$
By the First Isomorphism Theorem:
:$G / \map \ker \phi \cong \Img \phi$
Hence the result:
:$G / G \cong \set e$
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $G / G$ be the [[Definition:Quotient Group|quotient group]] of $G$ by itself.
Then:
:$G / G \cong \set e$
That is, the [[Definition:Quotient Group|quotient]] of a [[Definition:Group|group]] by itself is [[Definition:Group Isomorphism|isomorphic]] to the [[Definition:Triv... | Let the [[Definition:Group Homomorphism|homomorphism]] $\phi: G \to \set e$ be defined as:
:$\forall g \in G: \map \phi g = e$
Then:
:$\map \ker \phi = G$
and:
:$\Img \phi = \set e$
By the [[First Isomorphism Theorem]]:
:$G / \map \ker \phi \cong \Img \phi$
Hence the result:
:$G / G \cong \set e$
{{qed}} | Quotient of Group by Itself | https://proofwiki.org/wiki/Quotient_of_Group_by_Itself | https://proofwiki.org/wiki/Quotient_of_Group_by_Itself | [
"Examples of Quotient Groups"
] | [
"Definition:Group",
"Definition:Quotient Group",
"Definition:Quotient Group",
"Definition:Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Trivial Group"
] | [
"Definition:Group Homomorphism",
"First Isomorphism Theorem"
] |
proofwiki-13608 | Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999 | Let $n$ be an integer which has at least $3$ digits when expressed in decimal notation.
Let the digits of $n$ be divided into groups of $3$, counting from the right, and those groups added.
Then the result is equal to a multiple of $999$ {{iff}} $n$ is divisible by $999$. | {{refactor|The below sentence should be on the mistake page. Need to go back to the source work to clarify what we have.|level = medium}}
The mistake is either ''and conversely'' or ''equal to $999$'', since $999 \, 999$ is an easy counterexample.
Here we will show that the result is equal to '''a multiple of''' $999$ ... | Let $n$ be an [[Definition:Integer|integer]] which has at least $3$ [[Definition:Digit|digits]] when expressed in [[Definition:Decimal Notation|decimal notation]].
Let the [[Definition:Digit|digits]] of $n$ be divided into groups of $3$, counting from the right, and those groups added.
Then the result is equal to a ... | {{refactor|The below sentence should be on the mistake page. Need to go back to the source work to clarify what we have.|level = medium}}
The mistake is either ''and conversely'' or ''equal to $999$'', since $999 \, 999$ is an easy counterexample.
Here we will show that the result is equal to '''a multiple of''' $999... | Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999 | https://proofwiki.org/wiki/Integer_whose_Digits_when_Grouped_in_3s_add_to_Multiple_of_999_is_Divisible_by_999 | https://proofwiki.org/wiki/Integer_whose_Digits_when_Grouped_in_3s_add_to_Multiple_of_999_is_Divisible_by_999 | [
"Recreational Mathematics",
"Divisibility Tests",
"Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Decimal Notation",
"Definition:Digit",
"Definition:Multiple/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Congruence of Powers"
] |
proofwiki-13609 | Number which is Sum of Subfactorials of Digits | The only integer which is the sum of the subfactorials of its digits is $148 \, 349$:
:$148 \, 349 = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9$ | We have:
{{begin-eqn}}
{{eqn | l = 148 \, 349
| r = 0 + 9 + 14 \, 833 + 2 + 9 + 133 \, 496
| c =
}}
{{eqn | r = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9
| c =
}}
{{end-eqn}}
A computer search can verify solutions under $10^6$ (that is, with no more ... | The only [[Definition:Integer|integer]] which is the [[Definition:Integer Addition|sum]] of the [[Definition:Subfactorial|subfactorials]] of its [[Definition:Digit|digits]] is $148 \, 349$:
:$148 \, 349 = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9$ | We have:
{{begin-eqn}}
{{eqn | l = 148 \, 349
| r = 0 + 9 + 14 \, 833 + 2 + 9 + 133 \, 496
| c =
}}
{{eqn | r = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9
| c =
}}
{{end-eqn}}
A computer search can verify solutions under $10^6$ (that is, with no mo... | Number which is Sum of Subfactorials of Digits | https://proofwiki.org/wiki/Number_which_is_Sum_of_Subfactorials_of_Digits | https://proofwiki.org/wiki/Number_which_is_Sum_of_Subfactorials_of_Digits | [
"Subfactorials",
"148,349"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Subfactorial",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Addition/Integers",
"Definition:Subfactorial",
"Definition:Digit",
"Bernoulli's Inequality"
] |
proofwiki-13610 | Integers Representable as Product of both 3 and 4 Consecutive Integers | There are $3$ integers which can be expressed as both $x \paren {x + 1} \paren {x + 2} \paren {x + 3}$ for some $x$, and $y \paren {y + 1} \paren {y + 2}$ for some $y$:
:$24, 120, 175 \, 560$ | We have:
{{begin-eqn}}
{{eqn | l = 24
| r = 1 \times 2 \times 3 \times 4
| c =
}}
{{eqn | r = 2 \times 3 \times 4
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 120
| r = 2 \times 3 \times 4 \times 5
| c =
}}
{{eqn | r = 4 \times 5 \times 6
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eq... | There are $3$ [[Definition:Integer|integers]] which can be expressed as both $x \paren {x + 1} \paren {x + 2} \paren {x + 3}$ for some $x$, and $y \paren {y + 1} \paren {y + 2}$ for some $y$:
:$24, 120, 175 \, 560$ | We have:
{{begin-eqn}}
{{eqn | l = 24
| r = 1 \times 2 \times 3 \times 4
| c =
}}
{{eqn | r = 2 \times 3 \times 4
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 120
| r = 2 \times 3 \times 4 \times 5
| c =
}}
{{eqn | r = 4 \times 5 \times 6
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{... | Integers Representable as Product of both 3 and 4 Consecutive Integers | https://proofwiki.org/wiki/Integers_Representable_as_Product_of_both_3_and_4_Consecutive_Integers | https://proofwiki.org/wiki/Integers_Representable_as_Product_of_both_3_and_4_Consecutive_Integers | [
"Number Theory"
] | [
"Definition:Integer"
] | [] |
proofwiki-13611 | Squares whose Digits form Consecutive Increasing Integers | The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive increasing integers begins:
:$428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, \ldots$
{{OEIS|A030467}} | We have:
{{begin-eqn}}
{{eqn | l = 428^2
| r = 183 \, 184
| c =
}}
{{eqn | l = 573^2
| r = 328 \, 329
| c =
}}
{{eqn | l = 727^2
| r = 528 \, 529
| c =
}}
{{eqn | l = 846^2
| r = 715 \, 716
| c =
}}
{{eqn | l = 7810^2
| r = 6099 \, 6100
| c =
}}
{{eqn | l... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] whose [[Definition:Square (Algebra)|squares]] have a [[Definition:Decimal Notation|decimal representation]] consisting of the concatenation of $2$ consecutive increasing [[Definition:Integer|integers]] begins:
:$428, 573, 727, 846, 7810, 36... | We have:
{{begin-eqn}}
{{eqn | l = 428^2
| r = 183 \, 184
| c =
}}
{{eqn | l = 573^2
| r = 328 \, 329
| c =
}}
{{eqn | l = 727^2
| r = 528 \, 529
| c =
}}
{{eqn | l = 846^2
| r = 715 \, 716
| c =
}}
{{eqn | l = 7810^2
| r = 6099 \, 6100
| c =
}}
{{eqn | ... | Squares whose Digits form Consecutive Increasing Integers | https://proofwiki.org/wiki/Squares_whose_Digits_form_Consecutive_Increasing_Integers | https://proofwiki.org/wiki/Squares_whose_Digits_form_Consecutive_Increasing_Integers | [
"Square Numbers",
"Recreational Mathematics"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Square/Function",
"Definition:Decimal Notation",
"Definition:Integer"
] | [] |
proofwiki-13612 | Smallest Fifth Power which is Sum of 6 Fifth Powers | The smallest fifth power which is the sum of $6$ fifth powers is $12^5 = 248 \, 832$:
:$12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$ | We have:
{{begin-eqn}}
{{eqn | l = 12^5
| r = 248 \, 832
| c =
}}
{{eqn | r = 1024 + 3125 + 7776 + 16 \, 807 + 59 \, 049 + 161 \, 051
| c =
}}
{{eqn | r = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5
| c =
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown that this is the smallest.}} | The smallest [[Definition:Fifth Power|fifth power]] which is the [[Definition:Integer Addition|sum]] of $6$ [[Definition:Fifth Power|fifth powers]] is $12^5 = 248 \, 832$:
:$12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$ | We have:
{{begin-eqn}}
{{eqn | l = 12^5
| r = 248 \, 832
| c =
}}
{{eqn | r = 1024 + 3125 + 7776 + 16 \, 807 + 59 \, 049 + 161 \, 051
| c =
}}
{{eqn | r = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5
| c =
}}
{{end-eqn}}
{{ProofWanted|It remains to be shown that this is the smallest.}} | Smallest Fifth Power which is Sum of 6 Fifth Powers | https://proofwiki.org/wiki/Smallest_Fifth_Power_which_is_Sum_of_6_Fifth_Powers | https://proofwiki.org/wiki/Smallest_Fifth_Power_which_is_Sum_of_6_Fifth_Powers | [
"Fifth Powers"
] | [
"Definition:Fifth Power",
"Definition:Addition/Integers",
"Definition:Fifth Power"
] | [] |
proofwiki-13613 | Prime Numbers Embedded in Digits of Pi | The sequence of prime numbers that can be found starting from the beginning of the decimal expansion of $\pi$ (pi) begins:
:$3, 31, 314 \, 159, 31 \, 415 \, 926 \, 535 \, 897 \, 932 \, 384 \, 626 \, 433 \, 832 \, 795 \, 028 \, 841, \ldots$
{{OEIS|A005042}} | By inspection. | The [[Definition:Integer Sequence|sequence]] of [[Definition:Prime Number|prime numbers]] that can be found starting from the beginning of the [[Definition:Decimal Expansion|decimal expansion]] of [[Definition:Pi|$\pi$ (pi)]] begins:
:$3, 31, 314 \, 159, 31 \, 415 \, 926 \, 535 \, 897 \, 932 \, 384 \, 626 \, 433 \, 832... | By inspection. | Prime Numbers Embedded in Digits of Pi | https://proofwiki.org/wiki/Prime_Numbers_Embedded_in_Digits_of_Pi | https://proofwiki.org/wiki/Prime_Numbers_Embedded_in_Digits_of_Pi | [
"Pi",
"Prime Numbers"
] | [
"Definition:Integer Sequence",
"Definition:Prime Number",
"Definition:Decimal Expansion",
"Definition:Pi"
] | [] |
proofwiki-13614 | 333,667 is Unique Period Prime with Period 9 | $333 \, 667$ is a unique period prime whose reciprocal has a period of $9$:
:$\dfrac 1 {333 \, 667} = 0 \cdotp \dot 00000 \, 299 \dot 7$ | By long division:
<pre>
0.000002997000002...
---------------------
333667)1.000000000000000000
667334
--------
3326660
3003003
-------
3236570
3003003
-------
2335670
2335669
-------
1... | $333 \, 667$ is a [[Definition:Unique Period Prime|unique period prime]] whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $9$:
:$\dfrac 1 {333 \, 667} = 0 \cdotp \dot 00000 \, 299 \dot 7$ | By [[Definition:Long Division|long division]]:
<pre>
0.000002997000002...
---------------------
333667)1.000000000000000000
667334
--------
3326660
3003003
-------
3236570
3003003
-------
2335670
2335669
... | 333,667 is Unique Period Prime with Period 9 | https://proofwiki.org/wiki/333,667_is_Unique_Period_Prime_with_Period_9 | https://proofwiki.org/wiki/333,667_is_Unique_Period_Prime_with_Period_9 | [
"333,667",
"Examples of Unique Period Primes"
] | [
"Definition:Unique Period Prime",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Definition:Classical Algorithm/Division",
"Definition:Prime Number",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period",
"Period of Reciprocal of Prime",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Prime Number",
"Definition:Multiplicative Order of Integer",
"De... |
proofwiki-13615 | Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes | :$351 \, 120^3$ can be represented as the sum of $3$, $4$, $5$, $6$, $7$ or $8$ cubes. | {{begin-eqn}}
{{eqn | l = 351120^3
| r = 175560^3 + 234080^3 + 292600^3
}}
{{eqn | r = 2 \times 87780^3 + 204820^3 + 321860^3
}}
{{eqn | r = 2 \times 87780^3 + 175560^3 + 2 \times 263340^3
}}
{{eqn | r = 3 \times 117040^3 + 3 \times 234080^3
}}
{{eqn | r = 2 \times 58520^3 + 117040^3 + 3 \times 175560^3 + 292600^... | :$351 \, 120^3$ can be represented as the [[Definition:Integer Addition|sum]] of $3$, $4$, $5$, $6$, $7$ or $8$ [[Definition:Cube Number|cubes]]. | {{begin-eqn}}
{{eqn | l = 351120^3
| r = 175560^3 + 234080^3 + 292600^3
}}
{{eqn | r = 2 \times 87780^3 + 204820^3 + 321860^3
}}
{{eqn | r = 2 \times 87780^3 + 175560^3 + 2 \times 263340^3
}}
{{eqn | r = 3 \times 117040^3 + 3 \times 234080^3
}}
{{eqn | r = 2 \times 58520^3 + 117040^3 + 3 \times 175560^3 + 292600^... | Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes | https://proofwiki.org/wiki/Cube_which_can_be_Represented_as_Sum_of_3,_4,_5,_6,_7_or_8_Cubes | https://proofwiki.org/wiki/Cube_which_can_be_Represented_as_Sum_of_3,_4,_5,_6,_7_or_8_Cubes | [
"Cube Numbers",
"351,120"
] | [
"Definition:Addition/Integers",
"Definition:Cube Number"
] | [] |
proofwiki-13616 | Prime Gaps of 100 | The following pairs of consecutive prime numbers are those whose difference is $100$:
:$\tuple {396 \, 733, 396 \, 833}, \ldots$
{{expand|Only know the first pair so far. Research needed to find the next one(s).}} | Demonstrated by listing the prime gaps.
{{qed}} | The following [[Definition:Ordered Pair|pairs]] of consecutive [[Definition:Prime Number|prime numbers]] are those whose [[Definition:Integer Subtraction|difference]] is $100$:
:$\tuple {396 \, 733, 396 \, 833}, \ldots$
{{expand|Only know the first pair so far. Research needed to find the next one(s).}} | Demonstrated by listing the [[Definition:Prime Gap|prime gaps]].
{{qed}} | Prime Gaps of 100 | https://proofwiki.org/wiki/Prime_Gaps_of_100 | https://proofwiki.org/wiki/Prime_Gaps_of_100 | [
"Prime Gaps"
] | [
"Definition:Ordered Pair",
"Definition:Prime Number",
"Definition:Subtraction/Integers"
] | [
"Definition:Prime Gap"
] |
proofwiki-13617 | Property of 490,689 | The number $490 \, 689$ can be expressed as the sum of $3$ cubes in $2$ different ways:
:$490 \, 689 = 4^3 + 60^3 + 65^3 = 8^3 + 25^3 \times 78^3$
while at the same time the products of the contributory cube roots of each of those $2$ ways are equal:
:$4 \times 60 \times 65 = 8 \times 25 \times 78$ | {{begin-eqn}}
{{eqn | l = 490 \, 689
| r = 64 + 216 \, 000 + 274 \, 625
| c =
}}
{{eqn | r = 4^3 + 60^3 + 65^3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 490 \, 689
| r = 512 + 15 \, 625 + 474 \, 552
| c =
}}
{{eqn | r = 8^3 + 25^3 + 78^3
| c =
}}
{{end-eqn}}
Then:
{{begin-eq... | The number $490 \, 689$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Cube Number|cubes]] in $2$ different ways:
:$490 \, 689 = 4^3 + 60^3 + 65^3 = 8^3 + 25^3 \times 78^3$
while at the same time the [[Definition:Integer Multiplication|products]] of the contributory [[Definition:Cube R... | {{begin-eqn}}
{{eqn | l = 490 \, 689
| r = 64 + 216 \, 000 + 274 \, 625
| c =
}}
{{eqn | r = 4^3 + 60^3 + 65^3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 490 \, 689
| r = 512 + 15 \, 625 + 474 \, 552
| c =
}}
{{eqn | r = 8^3 + 25^3 + 78^3
| c =
}}
{{end-eqn}}
Then:
{{begin... | Property of 490,689 | https://proofwiki.org/wiki/Property_of_490,689 | https://proofwiki.org/wiki/Property_of_490,689 | [
"Cube Numbers",
"490,689"
] | [
"Definition:Addition/Integers",
"Definition:Cube Number",
"Definition:Multiplication/Integers",
"Definition:Cube Root"
] | [] |
proofwiki-13618 | 510,510 is Product of 4 Consecutive Fibonacci Numbers | $510 \, 510$ can be expressed as the product of $4$ distinct consecutive Fibonacci numbers:
:$510 \, 510 = 13 \times 21 \times 34 \times 55$
and is also the $7$th primorial:
:$510 \, 510 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$ | By observation:
{{begin-eqn}}
{{eqn | l = 510 \, 510
| r = 13 \times 21 \times 34 \times 55
| c =
}}
{{eqn | r = 13 \times \paren {3 \times 7} \times \paren {2 \times 17} \times \paren {5 \times 11}
| c =
}}
{{eqn | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17
| c =
}}
{{end... | $510 \, 510$ can be expressed as the [[Definition:Integer Multiplication|product]] of $4$ [[Definition:Distinct|distinct]] consecutive [[Definition:Fibonacci Number|Fibonacci numbers]]:
:$510 \, 510 = 13 \times 21 \times 34 \times 55$
and is also the $7$th [[Definition:Primorial|primorial]]:
:$510 \, 510 = 2 \times 3 \... | By observation:
{{begin-eqn}}
{{eqn | l = 510 \, 510
| r = 13 \times 21 \times 34 \times 55
| c =
}}
{{eqn | r = 13 \times \paren {3 \times 7} \times \paren {2 \times 17} \times \paren {5 \times 11}
| c =
}}
{{eqn | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17
| c =
}}
{{en... | 510,510 is Product of 4 Consecutive Fibonacci Numbers | https://proofwiki.org/wiki/510,510_is_Product_of_4_Consecutive_Fibonacci_Numbers | https://proofwiki.org/wiki/510,510_is_Product_of_4_Consecutive_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Primorials",
"510,510"
] | [
"Definition:Multiplication/Integers",
"Definition:Distinct",
"Definition:Fibonacci Number",
"Definition:Primorial"
] | [] |
proofwiki-13619 | Tableau Confutation contains Finite Tableau Confutation | Let $\mathbf H$ be a countable set of WFFs of propositional logic.
Let $T$ be a tableau confutation of $\mathbf H$.
Then there exists a finite rooted subtree of $T'$ that is also a tableau confutation of $\mathbf H'$. | For each node $v \in T$, let $\map p v$ be the path from $v$ to $r_T$, the root of $T$.
This path is unique by Path in Tree is Unique.
Let $\VV$ be the subtree of $T$ consisting those nodes $v$ of $T$ such that $\map p v$ is not contradictory.
{{AimForCont}} that $\VV$ were infinite.
Then by König's Tree Lemma, $\VV$ h... | Let $\mathbf H$ be a [[Definition:Countable Set|countable set]] of [[Definition:WFF of Propositional Logic|WFFs of propositional logic]].
Let $T$ be a [[Definition:Tableau Confutation|tableau confutation]] of $\mathbf H$.
Then there exists a [[Definition:Finite Tree|finite]] [[Definition:Rooted Subtree|rooted subtre... | For each [[Definition:Node of Tree|node]] $v \in T$, let $\map p v$ be the [[Definition:Path (Graph Theory)|path]] from $v$ to $r_T$, the [[Definition:Root Node|root]] of $T$.
This [[Definition:Path (Graph Theory)|path]] is unique by [[Path in Tree is Unique]].
Let $\VV$ be the [[Definition:Rooted Subtree|subtree]] ... | Tableau Confutation contains Finite Tableau Confutation | https://proofwiki.org/wiki/Tableau_Confutation_contains_Finite_Tableau_Confutation | https://proofwiki.org/wiki/Tableau_Confutation_contains_Finite_Tableau_Confutation | [
"Propositional Tableaux"
] | [
"Definition:Countable Set",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Tableau Confutation",
"Definition:Tree (Graph Theory)/Finite",
"Definition:Rooted Subtree",
"Definition:Tableau Confutation"
] | [
"Definition:Tree (Graph Theory)/Node",
"Definition:Path (Graph Theory)",
"Definition:Rooted Tree/Root Node",
"Definition:Path (Graph Theory)",
"Path in Tree is Unique",
"Definition:Rooted Subtree",
"Definition:Tree (Graph Theory)/Node",
"Definition:Contradictory/Branch",
"Definition:Infinite Set",
... |
proofwiki-13620 | Two-Sided Prime/Sequence | The complete sequence of two-sided primes is:
:$2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739 \, 397$ | {{ProofWanted}}
Category:Two-Sided Primes
ct1xp4yykscy4hdf7tndy9q4udrwcmo | The complete [[Definition:Integer Sequence|sequence]] of [[Definition:Two-Sided Prime|two-sided primes]] is:
:$2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739 \, 397$ | {{ProofWanted}}
[[Category:Two-Sided Primes]]
ct1xp4yykscy4hdf7tndy9q4udrwcmo | Two-Sided Prime/Sequence | https://proofwiki.org/wiki/Two-Sided_Prime/Sequence | https://proofwiki.org/wiki/Two-Sided_Prime/Sequence | [
"Two-Sided Primes"
] | [
"Definition:Integer Sequence",
"Definition:Two-Sided Prime"
] | [
"Category:Two-Sided Primes"
] |
proofwiki-13621 | Set of 7 Anagrams which are Square | The following integers are all anagrams, and all square:
:$1 \, 048 \, 576, 1 \, 056 \, 784, 1 \, 085 \, 764, 5 \, 740 \, 816, 5 \, 764 \, 801, 6 \, 754 \, 801, 7 \, 845 \, 601$ | {{begin-eqn}}
{{eqn | l = 1 \, 048 \, 576
| r = 1024^2
}}
{{eqn | l = 1 \, 056 \, 784
| r = 1028^2
}}
{{eqn | l = 1 \, 085 \, 764
| r = 1042^2
}}
{{eqn | l = 5 \, 740 \, 816
| r = 2396^2
}}
{{eqn | l = 5 \, 764 \, 801
| r = 2401^2
}}
{{eqn | l = 6 \, 754 \, 801
| r = 2599^2
}}
{{eqn ... | The following [[Definition:Integer|integers]] are all [[Definition:Anagram|anagrams]], and all [[Definition:Square Number|square]]:
:$1 \, 048 \, 576, 1 \, 056 \, 784, 1 \, 085 \, 764, 5 \, 740 \, 816, 5 \, 764 \, 801, 6 \, 754 \, 801, 7 \, 845 \, 601$ | {{begin-eqn}}
{{eqn | l = 1 \, 048 \, 576
| r = 1024^2
}}
{{eqn | l = 1 \, 056 \, 784
| r = 1028^2
}}
{{eqn | l = 1 \, 085 \, 764
| r = 1042^2
}}
{{eqn | l = 5 \, 740 \, 816
| r = 2396^2
}}
{{eqn | l = 5 \, 764 \, 801
| r = 2401^2
}}
{{eqn | l = 6 \, 754 \, 801
| r = 2599^2
}}
{{eqn ... | Set of 7 Anagrams which are Square | https://proofwiki.org/wiki/Set_of_7_Anagrams_which_are_Square | https://proofwiki.org/wiki/Set_of_7_Anagrams_which_are_Square | [
"Square Numbers"
] | [
"Definition:Integer",
"Definition:Anagram",
"Definition:Square Number"
] | [] |
proofwiki-13622 | Burnside's Lemma | Let $G$ be a finite group acting on a set $X$.
Let $X / G$ be the set of orbits under this action.
For $x \in X$, let $\Stab x$ be the stabilizer of $x$ by $G$.
For $g \in G$, let $X^g$ denotes the set of all elements of $X$ which are fixed by $g$:
:$X^g := \set {x \in X: g x = x}$
Then:
:$\size {X / G} = \dfrac 1 {\or... | {{begin-eqn}}
{{eqn | l = \frac 1 {\order G} \sum_{g \mathop \in G} \size {X^g}
| r = \frac 1 {\order G} \sum_{g \mathop \in G} \size {\set {x \in X: g x = x} }
| c = by definition
}}
{{eqn | r = \frac 1 {\order G} \sum_{x \mathop \in X} \size {\set {g \in G: g x = x} }
| c = Same summation, different... | Let $G$ be a [[Definition:Finite Group|finite group]] [[Definition:Group Action|acting]] on a [[Definition:Set|set]] $X$.
Let $X / G$ be the [[Definition:Set of Orbits|set of orbits]] under this action.
For $x \in X$, let $\Stab x$ be the [[Definition:Stabilizer|stabilizer]] of $x$ by $G$.
For $g \in G$, let $X^g$ d... | {{begin-eqn}}
{{eqn | l = \frac 1 {\order G} \sum_{g \mathop \in G} \size {X^g}
| r = \frac 1 {\order G} \sum_{g \mathop \in G} \size {\set {x \in X: g x = x} }
| c = by definition
}}
{{eqn | r = \frac 1 {\order G} \sum_{x \mathop \in X} \size {\set {g \in G: g x = x} }
| c = Same summation, different... | Burnside's Lemma | https://proofwiki.org/wiki/Burnside's_Lemma | https://proofwiki.org/wiki/Burnside's_Lemma | [
"Burnside's Lemma",
"Group Actions"
] | [
"Definition:Finite Group",
"Definition:Group Action",
"Definition:Set",
"Definition:Orbit (Group Theory)/Set of Orbits",
"Definition:Stabilizer",
"Definition:Set",
"Definition:Element",
"Definition:Fixed Element",
"Definition:Orbit (Group Theory)",
"Definition:Fixed Point"
] | [
"Orbit-Stabilizer Theorem"
] |
proofwiki-13623 | Smallest Cunningham Chain of the First Kind of Length 7 | The smallest Cunningham chain of the first kind of length $7$ is:
:$\left({1 \, 122 \, 659, 2 \, 245 \, 319, 4 \, 490 \, 639, 8 \, 981 \, 279, 17 \, 962 \, 559, 35 \, 925 \, 119, 71 \, 850 \, 239}\right)$ | Let $C$ denote the sequence in question.
We have that:
:$\dfrac {1 \, 122 \, 659 - 1} 2 = 561 \, 329 = 83 \times 6763$
and so is not prime.
Thus $1 \, 122 \, 659$ is not a safe prime, as is required for $C$ to be a Cunningham chain of the first kind.
Then:
{{begin-eqn}}
{{eqn | l = 2 \times 561 \, 329 + 1
| r = 1... | The smallest [[Definition:Cunningham Chain of the First Kind|Cunningham chain of the first kind]] of [[Definition:Length of Sequence|length]] $7$ is:
:$\left({1 \, 122 \, 659, 2 \, 245 \, 319, 4 \, 490 \, 639, 8 \, 981 \, 279, 17 \, 962 \, 559, 35 \, 925 \, 119, 71 \, 850 \, 239}\right)$ | Let $C$ denote the [[Definition:Sequence|sequence]] in question.
We have that:
:$\dfrac {1 \, 122 \, 659 - 1} 2 = 561 \, 329 = 83 \times 6763$
and so is not [[Definition:Prime Number|prime]].
Thus $1 \, 122 \, 659$ is not a [[Definition:Safe Prime|safe prime]], as is required for $C$ to be a [[Definition:Cunningham C... | Smallest Cunningham Chain of the First Kind of Length 7 | https://proofwiki.org/wiki/Smallest_Cunningham_Chain_of_the_First_Kind_of_Length_7 | https://proofwiki.org/wiki/Smallest_Cunningham_Chain_of_the_First_Kind_of_Length_7 | [
"Cunningham Chains"
] | [
"Definition:Cunningham Chain/First Kind",
"Definition:Length of Sequence"
] | [
"Definition:Sequence",
"Definition:Prime Number",
"Definition:Safe Prime",
"Definition:Cunningham Chain/First Kind",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime... |
proofwiki-13624 | Square of Repunit times Sum of Digits | The following pattern emerges:
{{begin-eqn}}
{{eqn | l = 121 \times \paren {1 + 2 + 1}
| r = 22^2
}}
{{eqn | l = 12 \, 321 \times \paren {1 + 2 + 3 + 2 + 1}
| r = 333^2
}}
{{eqn | l = 1 \, 234 \, 321 \times \paren {1 + 2 + 3 + 4 + 3 + 2 + 1}
| r = 4444^2
}}
{{end-eqn}}
and so on, up until $999 \, 999 ... | From Square of Repunit:
{{begin-eqn}}
{{eqn | l = 121
| r = 11^2
}}
{{eqn | l = 12 \, 321
| r = 111^2
}}
{{eqn | l = 1 \, 234 \, 321
| r = 1111^2
}}
{{end-eqn}}
and so on.
Then from 1+2+...+n+(n-1)+...+1 = n^2:
{{begin-eqn}}
{{eqn | l = 1 + 2 + 1
| r = 2^2
}}
{{eqn | l = 1 + 2 + 3 + 2 + 1
... | The following pattern emerges:
{{begin-eqn}}
{{eqn | l = 121 \times \paren {1 + 2 + 1}
| r = 22^2
}}
{{eqn | l = 12 \, 321 \times \paren {1 + 2 + 3 + 2 + 1}
| r = 333^2
}}
{{eqn | l = 1 \, 234 \, 321 \times \paren {1 + 2 + 3 + 4 + 3 + 2 + 1}
| r = 4444^2
}}
{{end-eqn}}
and so on, up until $999 \, 99... | From [[Square of Repunit]]:
{{begin-eqn}}
{{eqn | l = 121
| r = 11^2
}}
{{eqn | l = 12 \, 321
| r = 111^2
}}
{{eqn | l = 1 \, 234 \, 321
| r = 1111^2
}}
{{end-eqn}}
and so on.
Then from [[1+2+...+n+(n-1)+...+1 = n^2]]:
{{begin-eqn}}
{{eqn | l = 1 + 2 + 1
| r = 2^2
}}
{{eqn | l = 1 + 2 + 3 + 2... | Square of Repunit times Sum of Digits | https://proofwiki.org/wiki/Square_of_Repunit_times_Sum_of_Digits | https://proofwiki.org/wiki/Square_of_Repunit_times_Sum_of_Digits | [
"Repunits",
"Square Numbers"
] | [] | [
"Square of Repunit",
"1+2+...+n+(n-1)+...+1 = n^2"
] |
proofwiki-13625 | Factorisation of Quintic x^5 - x + n into Irreducible Quadratic and Irreducible Cubic | The quintic $x^5 - x + n$ can be factorized into the product of an irreducible quadratic and an an irreducible cubic {{iff}} $n$ is in the set:
:$\set {\pm 15, \pm 22 \, 440, \pm 2 \, 759 \, 640}$ | We have that:
{{begin-eqn}}
{{eqn | l = x^5 - x \pm 15
| r = \paren {x^2 \pm x + 3} \paren {x^3 \mp x^2 \mp 2 x \pm 5}
}}
{{eqn | l = x^5 - x \pm 22440
| r = \paren {x^2 \mp 12 x + 55} \paren {x^3 \pm 12 x^2 + 89 x \pm 408}
}}
{{eqn | l = x^5 - x \pm 2 \, 759 \, 640
| r = \paren {x^2 \pm 12 x + 377} \... | The [[Definition:Quintic Polynomial|quintic]] $x^5 - x + n$ can be [[Definition:Factorization|factorized]] into the [[Definition:Product of Polynomials|product]] of an [[Definition:Irreducible Polynomial|irreducible]] [[Definition:Quadratic Polynomial|quadratic]] and an an [[Definition:Irreducible Polynomial|irreducibl... | We have that:
{{begin-eqn}}
{{eqn | l = x^5 - x \pm 15
| r = \paren {x^2 \pm x + 3} \paren {x^3 \mp x^2 \mp 2 x \pm 5}
}}
{{eqn | l = x^5 - x \pm 22440
| r = \paren {x^2 \mp 12 x + 55} \paren {x^3 \pm 12 x^2 + 89 x \pm 408}
}}
{{eqn | l = x^5 - x \pm 2 \, 759 \, 640
| r = \paren {x^2 \pm 12 x + 377} \... | Factorisation of Quintic x^5 - x + n into Irreducible Quadratic and Irreducible Cubic | https://proofwiki.org/wiki/Factorisation_of_Quintic_x^5_-_x_+_n_into_Irreducible_Quadratic_and_Irreducible_Cubic | https://proofwiki.org/wiki/Factorisation_of_Quintic_x^5_-_x_+_n_into_Irreducible_Quadratic_and_Irreducible_Cubic | [
"Polynomial Theory"
] | [
"Definition:Quintic Polynomial",
"Definition:Divisor (Algebra)/Factorization",
"Definition:Multiplication of Polynomials",
"Definition:Irreducible Polynomial",
"Definition:Quadratic Polynomial",
"Definition:Irreducible Polynomial",
"Definition:Cubic Polynomial"
] | [] |
proofwiki-13626 | Factorial as Product of Consecutive Factorials | The only factorials which are the product of consecutive factorials are:
{{begin-eqn}}
{{eqn | l = 0!
| r = 0! \times 1!
| c =
}}
{{eqn | l = 1!
| r = 0! \times 1!
| c =
}}
{{eqn | l = 2!
| r = 1! \times 2!
| c =
}}
{{eqn | r = 0! \times 1! \times 2!
| c =
}}
{{eqn | l = 10... | Suppose $m, n \in \N$ and $m > n$.
Write $\map F {n, m} = n! \paren {n + 1}! \cdots m!$.
Suppose we have $\map F {n, m} > r!$ for some $r \in \N$.
Suppose further that there is a prime $p$ where $m < p \le r$.
We claim that $\map F {n, m}$ cannot be a factorial of any number.
{{AimForCont}} $\map F {n, m} = s!$ for som... | The only [[Definition:Factorial|factorials]] which are the product of consecutive [[Definition:Factorial|factorials]] are:
{{begin-eqn}}
{{eqn | l = 0!
| r = 0! \times 1!
| c =
}}
{{eqn | l = 1!
| r = 0! \times 1!
| c =
}}
{{eqn | l = 2!
| r = 1! \times 2!
| c =
}}
{{eqn | r = 0!... | Suppose $m, n \in \N$ and $m > n$.
Write $\map F {n, m} = n! \paren {n + 1}! \cdots m!$.
Suppose we have $\map F {n, m} > r!$ for some $r \in \N$.
Suppose further that there is a [[Definition:Prime Number|prime]] $p$ where $m < p \le r$.
We claim that $\map F {n, m}$ cannot be a [[Definition:Factorial|factorial]] ... | Factorial as Product of Consecutive Factorials | https://proofwiki.org/wiki/Factorial_as_Product_of_Consecutive_Factorials | https://proofwiki.org/wiki/Factorial_as_Product_of_Consecutive_Factorials | [
"Factorials",
"Factorial as Product of Consecutive Factorials"
] | [
"Definition:Factorial",
"Definition:Factorial"
] | [
"Definition:Prime Number",
"Definition:Factorial",
"Definition:Factorial",
"Definition:Lemma",
"Definition:Prime Number",
"Definition:Factorial",
"Definition:Prime Number",
"Definition:Factorial",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Fact... |
proofwiki-13627 | Numbers whose Fourth Root equals Number of Divisors | There are $4$ positive integers whose $4$th root equals the number of its divisors:
{{begin-eqn}}
{{eqn | l = 1
| r = 1^4
| c =
}}
{{eqn | l = 625
| r = 5^4
| c =
}}
{{eqn | l = 6561
| r = 9^4
| c =
}}
{{eqn | l = 4 \, 100 \, 625
| r = 45^4
| c =
}}
{{end-eqn}}
{{OEIS... | {{begin-eqn}}
{{eqn | l = \map {\sigma_0} 1
| r = 1
| c = {{DCFLink|1}}
}}
{{eqn | l = \map {\sigma_0} {625}
| r = 5
| c = {{DCFLink|625}}
}}
{{eqn | l = \map {\sigma_0} {6561}
| r = 9
| c = {{DCFLink|6561}}
}}
{{eqn | l = \map {\sigma_0} {4 \, 100 \, 625}
| r = 45
| c = ... | There are $4$ [[Definition:Positive Integer|positive integers]] whose [[Definition:Root of Number|$4$th root]] equals the number of its [[Definition:Divisor of Integer|divisors]]:
{{begin-eqn}}
{{eqn | l = 1
| r = 1^4
| c =
}}
{{eqn | l = 625
| r = 5^4
| c =
}}
{{eqn | l = 6561
| r = 9^... | {{begin-eqn}}
{{eqn | l = \map {\sigma_0} 1
| r = 1
| c = {{DCFLink|1}}
}}
{{eqn | l = \map {\sigma_0} {625}
| r = 5
| c = {{DCFLink|625}}
}}
{{eqn | l = \map {\sigma_0} {6561}
| r = 9
| c = {{DCFLink|6561}}
}}
{{eqn | l = \map {\sigma_0} {4 \, 100 \, 625}
| r = 45
| c = ... | Numbers whose Fourth Root equals Number of Divisors | https://proofwiki.org/wiki/Numbers_whose_Fourth_Root_equals_Number_of_Divisors | https://proofwiki.org/wiki/Numbers_whose_Fourth_Root_equals_Number_of_Divisors | [
"Fourth Powers",
"Divisor Count Function"
] | [
"Definition:Positive/Integer",
"Definition:Root of Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Count Function is Odd Iff Argument is Square",
"Definition:Odd Integer",
"Definition:Prime Power",
"Divisor Count Function of Power of Prime",
"Bernoulli's Inequality",
"Definition:Odd Prime",
"Definition:Prime Power",
"Divisor Count Function is Multiplicative",
"Definition:Integer",
"Def... |
proofwiki-13628 | Weak Existence of Matrix Logarithm | Let $T$ be a square matrix of order $n$.
Let $\norm {T - I} < 1$ in the norm on bounded linear operators, where $I$ the identity matrix.
Then there is a square matrix $S$ such that:
:$e^S = T$
where $e^S$ is the matrix exponential. | Define:
:$\ds S = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \paren {T - I}^n$
$S$ converges since $\norm {T - I} < 1$.
We have that $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \norm {T - I}^n$ is the Newton-Mercator Series.
This converges since $\norm {T - I} < 1$.
Hence the series ... | Let $T$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\norm {T - I} < 1$ in the [[Definition:Norm on Bounded Linear Transformation|norm on bounded linear operators]], where $I$ the [[Definition:Identity Matrix|identity matrix]].
Then there is a [[Definition:... | Define:
:$\ds S = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \paren {T - I}^n$
$S$ converges since $\norm {T - I} < 1$.
We have that $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \norm {T - I}^n$ is the [[Definition:Newton-Mercator Series|Newton-Mercator Series]].
This converges si... | Weak Existence of Matrix Logarithm | https://proofwiki.org/wiki/Weak_Existence_of_Matrix_Logarithm | https://proofwiki.org/wiki/Weak_Existence_of_Matrix_Logarithm | [
"Matrix Algebra",
"Matrix Logarithms"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix Exponential"
] | [
"Definition:Newton-Mercator Series",
"Properties of Matrix Exponential",
"Definition:Newton-Mercator Series",
"Power Series Expansion for Exponential Function"
] |
proofwiki-13629 | Norm on Vector Space is Continuous Function | Let $V$ be a vector space with norm $\norm {\, \cdot \,}$.
The function $\norm {\, \cdot \,}: V \to \R$ is continuous. | Let $x_n \to x$ in $V$.
Then we have:
:$\norm {x_n - x} \to 0$
By the Reverse Triangle Inequality:
:$\size {\norm {x_n} - \norm x} \le \norm {x_n - x}$
Hence:
:$\size {\norm {x_n} - \norm x} \to 0$
Thus:
:$\norm {x_n} \to \norm x$
Hence the result from the definition of a continuous real function.
{{qed}}
Category:Norm... | Let $V$ be a vector space with norm $\norm {\, \cdot \,}$.
The function $\norm {\, \cdot \,}: V \to \R$ is continuous. | Let $x_n \to x$ in $V$.
Then we have:
:$\norm {x_n - x} \to 0$
By the [[Reverse Triangle Inequality]]:
:$\size {\norm {x_n} - \norm x} \le \norm {x_n - x}$
Hence:
:$\size {\norm {x_n} - \norm x} \to 0$
Thus:
:$\norm {x_n} \to \norm x$
Hence the result from the definition of a [[Definition:Continuous Real Function ... | Norm on Vector Space is Continuous Function | https://proofwiki.org/wiki/Norm_on_Vector_Space_is_Continuous_Function | https://proofwiki.org/wiki/Norm_on_Vector_Space_is_Continuous_Function | [
"Norm Theory"
] | [] | [
"Reverse Triangle Inequality",
"Definition:Continuous Real Function/Point",
"Category:Norm Theory"
] |
proofwiki-13630 | Smallest 5 Consecutive Primes in Arithmetic Sequence | The smallest $5$ consecutive primes in arithmetic sequence are:
:$9 \, 843 \, 019 + 30 n$
for $n = 0, 1, 2, 3, 4$.
Note that while there are many longer arithmetic sequences of far smaller primes, those primes are not consecutive. | {{begin-eqn}}
{{eqn | l = 9 \, 843 \, 019 + 0 \times 30
| r = 9 \, 843 \, 019
| c = which is the $654 \, 926$th prime
}}
{{eqn | l = 9 \, 843 \, 019 + 1 \times 30
| r = 9 \, 843 \, 049
| c = which is the $654 \, 927$th prime
}}
{{eqn | l = 9 \, 843 \, 019 + 2 \times 30
| r = 9 \, 843 \, 07... | The smallest $5$ consecutive [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] are:
:$9 \, 843 \, 019 + 30 n$
for $n = 0, 1, 2, 3, 4$.
Note that while there are many longer [[Definition:Arithmetic Sequence|arithmetic sequences]] of far smaller [[Definition:Prime Number|prime... | {{begin-eqn}}
{{eqn | l = 9 \, 843 \, 019 + 0 \times 30
| r = 9 \, 843 \, 019
| c = which is the $654 \, 926$th [[Definition:Prime Number|prime]]
}}
{{eqn | l = 9 \, 843 \, 019 + 1 \times 30
| r = 9 \, 843 \, 049
| c = which is the $654 \, 927$th [[Definition:Prime Number|prime]]
}}
{{eqn | l = ... | Smallest 5 Consecutive Primes in Arithmetic Sequence | https://proofwiki.org/wiki/Smallest_5_Consecutive_Primes_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Smallest_5_Consecutive_Primes_in_Arithmetic_Sequence | [
"Prime Numbers",
"Arithmetic Sequences"
] | [
"Definition:Prime Number",
"Definition:Arithmetic Sequence",
"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"
] |
proofwiki-13631 | Number of Ways to Tile Standard Chessboard with Dominoes | The number of ways to tile a standard ($8 \times 8$) chessboard with dominoes is $12 \, 988 \, 816$. | Follows directly from Kasteleyn's Formula for $m = n = 8$:
:$\ds \prod_{j \mathop = 1}^{\ceiling {\frac 8 2} } \prod_{k \mathop = 1}^{\ceiling {\frac 8 2} } \paren {4 \cos^2 \frac {\pi j} {8 + 1} + 4 \cos^2 \frac {\pi k} {8 + 1} } = 12 \, 988 \, 816$
{{qed}}
{{explain|The above really needs to be evaluated step by step... | The number of ways to [[Definition:Chessboard Tiling|tile]] a [[Definition:Chessboard|standard ($8 \times 8$) chessboard]] with [[Definition:Domino|dominoes]] is $12 \, 988 \, 816$. | Follows directly from [[Kasteleyn's Formula]] for $m = n = 8$:
:$\ds \prod_{j \mathop = 1}^{\ceiling {\frac 8 2} } \prod_{k \mathop = 1}^{\ceiling {\frac 8 2} } \paren {4 \cos^2 \frac {\pi j} {8 + 1} + 4 \cos^2 \frac {\pi k} {8 + 1} } = 12 \, 988 \, 816$
{{qed}}
{{explain|The above really needs to be evaluated step b... | Number of Ways to Tile Standard Chessboard with Dominoes | https://proofwiki.org/wiki/Number_of_Ways_to_Tile_Standard_Chessboard_with_Dominoes | https://proofwiki.org/wiki/Number_of_Ways_to_Tile_Standard_Chessboard_with_Dominoes | [
"Chessboard Tilings",
"Chessboard Puzzles",
"Dominoes",
"Recreational Mathematics"
] | [
"Definition:Chessboard Tiling",
"Definition:Chess/Chessboard",
"Definition:Domino"
] | [
"Kasteleyn's Formula"
] |
proofwiki-13632 | Smallest Triplet of Primitive Pythagorean Triangles with Same Area | The smallest set of $3$ primitive Pythagorean triangles which all have the same area are:
:the $4485-5852-7373$ triangle
:the $3059-8580-9109$ triangle
:the $1380-19 \, 019-19 \, 069$ triangle.
That area is $13 \, 123 \, 110$. | We have that:
:the $4485-5852-7373$ triangle $T_1$ is Pythagorean
:the $3059-8580-9109$ triangle $T_2$ is Pythagorean
:the $1380-19 \, 019-19 \, 069$ triangle $T_3$ is Pythagorean.
Then from Area of Triangle, their areas $A_1$, $A_2$ and $A_3$ respectively are given by:
{{begin-eqn}}
{{eqn | l = A_1
| r = \dfrac ... | The smallest [[Definition:Set|set of $3$]] [[Definition:Primitive Pythagorean Triangle|primitive Pythagorean triangles]] which all have the same [[Definition:Area|area]] are:
:the [[Pythagorean Triangle/Examples/4485-5852-7373|$4485-5852-7373$ triangle]]
:the [[Pythagorean Triangle/Examples/3059-8580-9109|$3059-8580-... | We have that:
:the [[Pythagorean Triangle/Examples/4485-5852-7373|$4485-5852-7373$ triangle $T_1$ is Pythagorean]]
:the [[Pythagorean Triangle/Examples/3059-8580-9109|$3059-8580-9109$ triangle $T_2$ is Pythagorean]]
:the [[Pythagorean Triangle/Examples/1380-19,019-19,069|$1380-19 \, 019-19 \, 069$ triangle $T_3$ is Pyt... | Smallest Triplet of Primitive Pythagorean Triangles with Same Area | https://proofwiki.org/wiki/Smallest_Triplet_of_Primitive_Pythagorean_Triangles_with_Same_Area | https://proofwiki.org/wiki/Smallest_Triplet_of_Primitive_Pythagorean_Triangles_with_Same_Area | [
"Specific Numbers",
"13,123,110"
] | [
"Definition:Set",
"Definition:Primitive Pythagorean Triangle",
"Definition:Area",
"Pythagorean Triangle/Examples/4485-5852-7373",
"Pythagorean Triangle/Examples/3059-8580-9109",
"Pythagorean Triangle/Examples/1380-19,019-19,069"
] | [
"Pythagorean Triangle/Examples/4485-5852-7373",
"Pythagorean Triangle/Examples/3059-8580-9109",
"Pythagorean Triangle/Examples/1380-19,019-19,069",
"Area of Triangle",
"Definition:Area"
] |
proofwiki-13633 | Smallest Even Integer whose Euler Phi Value is not the Euler Phi Value of an Odd Integer | The smallest even integer whose Euler $\phi$ value is shared by no odd integer is $33 \, 817 \, 088$. | We have:
{{begin-eqn}}
{{eqn | l = \map \phi {33 \, 817 \, 088}
| r = 16 \, 842 \, 752
| c = {{EulerPhiLink|33,817,088|33 \, 817 \, 088}}
}}
{{eqn | r = 2^{16} \times 257
| c =
}}
{{end-eqn}}
Consider the equation:
:$(1): \quad \map \phi x = 2^{16} \times 257$
Let $p$ be an odd prime factor of $x$.
T... | The smallest [[Definition:Even Integer|even integer]] whose [[Definition:Euler Phi Function|Euler $\phi$ value]] is shared by no [[Definition:Odd Integer|odd integer]] is $33 \, 817 \, 088$. | We have:
{{begin-eqn}}
{{eqn | l = \map \phi {33 \, 817 \, 088}
| r = 16 \, 842 \, 752
| c = {{EulerPhiLink|33,817,088|33 \, 817 \, 088}}
}}
{{eqn | r = 2^{16} \times 257
| c =
}}
{{end-eqn}}
Consider the equation:
:$(1): \quad \map \phi x = 2^{16} \times 257$
Let $p$ be an [[Definition:Odd Intege... | Smallest Even Integer whose Euler Phi Value is not the Euler Phi Value of an Odd Integer | https://proofwiki.org/wiki/Smallest_Even_Integer_whose_Euler_Phi_Value_is_not_the_Euler_Phi_Value_of_an_Odd_Integer | https://proofwiki.org/wiki/Smallest_Even_Integer_whose_Euler_Phi_Value_is_not_the_Euler_Phi_Value_of_an_Odd_Integer | [
"33,817,088",
"Euler Phi Function"
] | [
"Definition:Even Integer",
"Definition:Euler Phi Function",
"Definition:Odd Integer"
] | [
"Definition:Odd Integer",
"Definition:Prime Factor",
"Euler Phi Function is Multiplicative",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fermat Prime",
"Definition:Composite Number",
"Definition:Odd Integer",
"De... |
proofwiki-13634 | Squares whose Digits form Consecutive Integers | The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive integers, either increasing or decreasing begins:
:$91, 428, 573, 727, 846, 7810, 9079, 9901, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, 733 \, 674, \ldots$
This sequence can be divided into two su... | We have:
{{begin-eqn}}
{{eqn | l = 91^2
| r = 8281
| c =
}}
{{eqn | l = 428^2
| r = 183 \, 184
| c =
}}
{{eqn | l = 573^2
| r = 328 \, 329
| c =
}}
{{eqn | l = 727^2
| r = 528 \, 529
| c =
}}
{{eqn | l = 846^2
| r = 715 \, 716
| c =
}}
{{eqn | l = 7810^2
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] whose [[Definition:Square (Algebra)|squares]] have a [[Definition:Decimal Notation|decimal representation]] consisting of the concatenation of $2$ consecutive [[Definition:Integer|integers]], either increasing or decreasing begins:
:$91, 42... | We have:
{{begin-eqn}}
{{eqn | l = 91^2
| r = 8281
| c =
}}
{{eqn | l = 428^2
| r = 183 \, 184
| c =
}}
{{eqn | l = 573^2
| r = 328 \, 329
| c =
}}
{{eqn | l = 727^2
| r = 528 \, 529
| c =
}}
{{eqn | l = 846^2
| r = 715 \, 716
| c =
}}
{{eqn | l = 7810^2... | Squares whose Digits form Consecutive Integers | https://proofwiki.org/wiki/Squares_whose_Digits_form_Consecutive_Integers | https://proofwiki.org/wiki/Squares_whose_Digits_form_Consecutive_Integers | [
"Square Numbers",
"Recreational Mathematics"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Square/Function",
"Definition:Decimal Notation",
"Definition:Integer",
"Definition:Integer",
"Definition:Integer"
] | [] |
proofwiki-13635 | Squares whose Digits form Consecutive Decreasing Integers | The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers begins:
:$91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, 99 \, 990 \, 001, \ldots$
{{OEIS|A030467}} | We have:
{{begin-eqn}}
{{eqn | l = 91^2
| r = 8281
| c =
}}
{{eqn | l = 9079^2
| r = 82 \, 428 \, 241
| c =
}}
{{eqn | l = 9901^2
| r = 98 \, 029 \, 801
| c =
}}
{{eqn | l = 733 \, 674^2
| r = 538 \, 277 \, 538 \, 276
| c =
}}
{{eqn | l = 999 \, 001^2
| r = 998 ... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Integer|integers]] whose [[Definition:Square (Algebra)|squares]] have a [[Definition:Decimal Notation|decimal representation]] consisting of the concatenation of $2$ consecutive decreasing [[Definition:Integer|integers]] begins:
:$91, 9079, 9901, 733 \, 674, ... | We have:
{{begin-eqn}}
{{eqn | l = 91^2
| r = 8281
| c =
}}
{{eqn | l = 9079^2
| r = 82 \, 428 \, 241
| c =
}}
{{eqn | l = 9901^2
| r = 98 \, 029 \, 801
| c =
}}
{{eqn | l = 733 \, 674^2
| r = 538 \, 277 \, 538 \, 276
| c =
}}
{{eqn | l = 999 \, 001^2
| r = 998... | Squares whose Digits form Consecutive Decreasing Integers | https://proofwiki.org/wiki/Squares_whose_Digits_form_Consecutive_Decreasing_Integers | https://proofwiki.org/wiki/Squares_whose_Digits_form_Consecutive_Decreasing_Integers | [
"Square Numbers",
"Recreational Mathematics"
] | [
"Definition:Integer Sequence",
"Definition:Integer",
"Definition:Square/Function",
"Definition:Decimal Notation",
"Definition:Integer"
] | [] |
proofwiki-13636 | Numbers n whose Euler Phi value Divides n + 1 | The following integers $n$ satisfy the equation:
:$\exists k \in \Z: k \, \map \phi n = n + 1$
where $\phi$ denotes the Euler $\phi$ function:
:$83 \, 623 \, 935, 83 \, 623 \, 935 \times 83 \, 623 \, 937$ | From {{EulerPhiLink|83,623,935|83 \, 623 \, 935}}:
:$\map \phi {83 \, 623 \, 935} = 41 \, 811 \, 968$
and then:
{{begin-eqn}}
{{eqn | l = 2 \times 41 \, 811 \, 968
| r = 83 \, 623 \, 936
| c =
}}
{{eqn | r = 1 + 83 \, 623 \, 935
| c =
}}
{{end-eqn}}
{{qed|lemma}}
Then we have that $83 \, 623 \, 937$... | The following [[Definition:Integer|integers]] $n$ satisfy the equation:
:$\exists k \in \Z: k \, \map \phi n = n + 1$
where $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]]:
:$83 \, 623 \, 935, 83 \, 623 \, 935 \times 83 \, 623 \, 937$ | From {{EulerPhiLink|83,623,935|83 \, 623 \, 935}}:
:$\map \phi {83 \, 623 \, 935} = 41 \, 811 \, 968$
and then:
{{begin-eqn}}
{{eqn | l = 2 \times 41 \, 811 \, 968
| r = 83 \, 623 \, 936
| c =
}}
{{eqn | r = 1 + 83 \, 623 \, 935
| c =
}}
{{end-eqn}}
{{qed|lemma}}
Then we have that $83 \, 623 \, ... | Numbers n whose Euler Phi value Divides n + 1 | https://proofwiki.org/wiki/Numbers_n_whose_Euler_Phi_value_Divides_n_+_1 | https://proofwiki.org/wiki/Numbers_n_whose_Euler_Phi_value_Divides_n_+_1 | [
"Euler Phi Function"
] | [
"Definition:Integer",
"Definition:Euler Phi Function"
] | [
"Definition:Prime Number",
"Euler Phi Function of Prime",
"Euler Phi Function is Multiplicative",
"Difference of Two Squares"
] |
proofwiki-13637 | Polynomial is Linear Combination of Monomials | Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be a polynomial ring over $R$ in the variable $X$.
Let $P \in R \sqbrk X$.
Then $P$ is a linear combination of the monomials of $R \sqbrk X$, with coefficients in $R$.
{{explain|this needs to be made more precise}} | Let $S \subset R \sqbrk X$ be the subset of all elements that are linear combinations of monomials.
Let $\iota: S \to R \sqbrk X$ denote the inclusion mapping.
Suppose for the moment that $S$ is a commutative ring with unity.
Then by Universal Property of Polynomial Ring, there exists a ring homomorphism $g: R \sqbrk X... | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $R \sqbrk X$ be a [[Definition:Polynomial Ring|polynomial ring]] over $R$ in the [[Definition:Variable of Polynomial Ring|variable]] $X$.
Let $P \in R \sqbrk X$.
Then $P$ is a [[Definition:Linear Combination|linear combination]... | Let $S \subset R \sqbrk X$ be the [[Definition:Subset|subset]] of all elements that are [[Definition:Linear Combination|linear combinations]] of [[Definition:Monomial of Polynomial Ring|monomials]].
Let $\iota: S \to R \sqbrk X$ denote the [[Definition:Inclusion Mapping|inclusion mapping]].
Suppose for the moment tha... | Polynomial is Linear Combination of Monomials | https://proofwiki.org/wiki/Polynomial_is_Linear_Combination_of_Monomials | https://proofwiki.org/wiki/Polynomial_is_Linear_Combination_of_Monomials | [
"Polynomial Theory",
"Monomials"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Linear Combination",
"Definition:Monomial of Polynomial Ring"
] | [
"Definition:Subset",
"Definition:Linear Combination",
"Definition:Monomial of Polynomial Ring",
"Definition:Inclusion Mapping",
"Definition:Commutative and Unitary Ring",
"Universal Property of Polynomial Ring",
"Definition:Ring Homomorphism",
"Inclusion Mapping on Subring is Homomorphism",
"Definit... |
proofwiki-13638 | Sum over Disjoint Union of Finite Sets | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ and $T$ be finite disjoint sets.
Let $S \cup T$ be their union.
Let $f: S \cup T \to \mathbb A$ be a mapping.
Then we have the equality of summations over finite sets:
:$\ds \sum_{u \mathop \in S \mathop \cup T} \map f u = \sum_{s \math... | Note that by Union of Finite Sets is Finite, the union $S \cup T$ is finite.
Let $m$ be the cardinality of $S$ and $n$ be the cardinality of $T$.
Let $\N_{< m}$ denote an initial segment of the natural numbers.
Let $\sigma: \N_{<m} \to S$ and $\tau: \N_{<n} \to T$ be bijections.
Let $\alpha: \N_{< n} \to \closedint m {... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ and $T$ be [[Definition:Finite Set|finite]] [[Definition:Disjoint Sets|disjoint sets]].
Let $S \cup T$ be their [[Definition:Set Union|union]].
Let $f: S \cup T \to \mathbb A$ be a [[Definition:M... | Note that by [[Union of Finite Sets is Finite]], the [[Definition:Set Union|union]] $S \cup T$ is [[Definition:Finite Set|finite]].
Let $m$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$ and $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $T$.
Let $\N_{< m}$ denote an [[Definiti... | Sum over Disjoint Union of Finite Sets | https://proofwiki.org/wiki/Sum_over_Disjoint_Union_of_Finite_Sets | https://proofwiki.org/wiki/Sum_over_Disjoint_Union_of_Finite_Sets | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Disjoint Sets",
"Definition:Set Union",
"Definition:Mapping",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Union of Finite Sets is Finite",
"Definition:Set Union",
"Definition:Finite Set",
"Definition:Cardinality/Finite",
"Definition:Cardinality/Finite",
"Definition:Initial Segment of Natural Numbers",
"Definition:Bijection",
"Definition:Mapping",
"Translation of Integer Interval is Bijection",
"Defin... |
proofwiki-13639 | Finite Summation does not Change under Permutation | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f: S \to \mathbb A$ be a mapping.
Let $\sigma: S \to S$ be a permutation.
Then we have the equality of summations over finite sets:
:$\ds \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in S} \map f {\map \sigma... | This is a special case of Change of Variables in Summation over Finite Set.
{{qed}}
Category:Summations
glofy0i42m26o45ix2z326ih5804vid | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Let $\sigma: S \to S$ be a [[Definition:Permutation|permutation]].
Then we have the equa... | This is a special case of [[Change of Variables in Summation over Finite Set]].
{{qed}}
[[Category:Summations]]
glofy0i42m26o45ix2z326ih5804vid | Finite Summation does not Change under Permutation | https://proofwiki.org/wiki/Finite_Summation_does_not_Change_under_Permutation | https://proofwiki.org/wiki/Finite_Summation_does_not_Change_under_Permutation | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Permutation",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Change of Variables in Summation over Finite Set",
"Category:Summations"
] |
proofwiki-13640 | Summation over Finite Set is Well-Defined | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f: S \to \mathbb A$ be a mapping.
Let $n$ be the cardinality of $S$.
let $\N_{<n}$ be an initial segment of the natural numbers.
Let $g, h: \N_{<n} \to S$ be bijections.
Then we have an equality of indexed summatio... | By Inverse of Bijection is Bijection, $h^{-1} : \N_{<n} \to S$ is a bijection.
By Composite of Bijections is Bijection, the composition $h^{-1}\circ g$ is a permutation of $\N_{<n}$.
By Indexed Summation does not Change under Permutation, we have an equality of indexed summations:
:$\ds \sum_{i \mathop = 0}^{n - 1} \ma... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$.
let $\N_{<n}$... | By [[Inverse of Bijection is Bijection]], $h^{-1} : \N_{<n} \to S$ is a [[Definition:Bijection|bijection]].
By [[Composite of Bijections is Bijection]], the [[Definition:Composition of Mappings|composition]] $h^{-1}\circ g$ is a [[Definition:Permutation|permutation]] of $\N_{<n}$.
By [[Indexed Summation does not Chan... | Summation over Finite Set is Well-Defined | https://proofwiki.org/wiki/Summation_over_Finite_Set_is_Well-Defined | https://proofwiki.org/wiki/Summation_over_Finite_Set_is_Well-Defined | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Cardinality/Finite",
"Definition:Initial Segment of Natural Numbers",
"Definition:Bijection",
"Definition:Summation/Indexed",
"Definition:Composition of Mappings",
"Definition:Summation",
"Definition:Finite Set",
"De... | [
"Inverse of Bijection is Bijection",
"Definition:Bijection",
"Composite of Bijections is Bijection",
"Definition:Composition of Mappings",
"Definition:Permutation",
"Indexed Summation does not Change under Permutation",
"Definition:Summation/Indexed",
"Composition of Mappings is Associative",
"Compo... |
proofwiki-13641 | Change of Variables in Summation over Finite Set | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ and $T$ be finite sets.
Let $f: S \to \mathbb A$ be a mapping.
Let $g: T \to S$ be a bijection.
Then we have an equality of summations over finite sets:
:$\ds \sum_{s \mathop \in S} \map f s = \sum_{t \mathop \in T} \map f {\map g t}$ | Let $n$ be the cardinality of $S$ and $T$.
Let $\N_{<n}$ be an initial segment of the natural numbers.
Let $h : \N_{<n} \to T$ be a bijection.
By definition of summation:
:$\ds \sum_{t \mathop \in T} \map f {\map g t} = \sum_{i \mathop = 0}^{n - 1} \map f {\map g {\map h i} }$
By Composite of Bijections is Bijection, t... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ and $T$ be [[Definition:Finite Set|finite sets]].
Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Let $g: T \to S$ be a [[Definition:Bijection|bijection]].
Then we have an equalit... | Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$ and $T$.
Let $\N_{<n}$ be an [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]].
Let $h : \N_{<n} \to T$ be a [[Definition:Bijection|bijection]].
By definition of [[Definition:Summation|summation]]:
:$\ds... | Change of Variables in Summation over Finite Set | https://proofwiki.org/wiki/Change_of_Variables_in_Summation_over_Finite_Set | https://proofwiki.org/wiki/Change_of_Variables_in_Summation_over_Finite_Set | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Definition:Cardinality/Finite",
"Definition:Initial Segment of Natural Numbers",
"Definition:Bijection",
"Definition:Summation",
"Composite of Bijections is Bijection",
"Definition:Composition of Mappings",
"Definition:Bijection",
"Definition:Summation"
] |
proofwiki-13642 | Indexed Summation does not Change under Permutation | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a$ and $b$ be integers.
Let $\closedint a b$ be the integer interval between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a mapping.
Let $\sigma: \closedint a b \to \closedint a b$ be a permutation.
Then we have an equality of ind... | Let $a > b$.
Then by definition of indexed summation, both sides are $0$.
Let $a \le b$. | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a$ and $b$ be [[Definition:Integer|integers]].
Let $\closedint a b$ be the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a [[Definiti... | Let $a > b$.
Then by definition of [[Definition:Indexed Summation|indexed summation]], both sides are $0$.
Let $a \le b$. | Indexed Summation does not Change under Permutation | https://proofwiki.org/wiki/Indexed_Summation_does_not_Change_under_Permutation | https://proofwiki.org/wiki/Indexed_Summation_does_not_Change_under_Permutation | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Permutation",
"Definition:Summation/Indexed"
] | [
"Definition:Summation/Indexed"
] |
proofwiki-13643 | Indexed Summation over Translated Interval | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a$ and $b$ be integers.
Let $\closedint a b$ be the integer interval between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a mapping.
Let $c\in\Z$ be an integer.
Then we have an equality of indexed summations:
:$\ds \sum_{i \mathop... | The proof goes by induction on $b$. | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a$ and $b$ be [[Definition:Integer|integers]].
Let $\closedint a b$ be the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a [[Definiti... | The proof goes by [[Principle of Mathematical Induction|induction]] on $b$. | Indexed Summation over Translated Interval | https://proofwiki.org/wiki/Indexed_Summation_over_Translated_Interval | https://proofwiki.org/wiki/Indexed_Summation_over_Translated_Interval | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Integer",
"Definition:Summation/Indexed"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-13644 | Indexed Summation over Adjacent Intervals | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a, b, c$ be integers.
Let $\closedint a c$ denote the integer interval between $a$ and $c$.
Let $b \in \closedint {a - 1} c$.
Let $f : \closedint a c \to \mathbb A$ be a mapping.
Then we have an equality of indexed summations:
:$\ds \sum_{... | The proof goes by induction on $b$. | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a, b, c$ be [[Definition:Integer|integers]].
Let $\closedint a c$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $c$.
Let $b \in \closedint {a - 1} c$.
Let $f : \closed... | The proof goes by [[Principle of Mathematical Induction|induction]] on $b$. | Indexed Summation over Adjacent Intervals | https://proofwiki.org/wiki/Indexed_Summation_over_Adjacent_Intervals | https://proofwiki.org/wiki/Indexed_Summation_over_Adjacent_Intervals | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Summation/Indexed"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-13645 | Indexed Summation over Interval of Length Two | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a \in \Z$ be an integer.
Let $f: \set {a, a + 1} \to \mathbb A$ be a real-valued function.
Then the indexed summation:
:$\ds \sum_{i \mathop = a}^{a + 1} \map f i = \map f a + \map f {a + 1}$ | We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = a}^{a + 1} \map f i
| r = \sum_{i \mathop = a}^a \map f i + \map f {a + 1}
| c = {{Defof|Indexed Summation}}
}}
{{eqn | l =
| r = \map f a + \map f {a + 1}
| c = Indexed Summation over Interval of Length One
}}
{{end-eqn}}
{{qed}}
Category:Su... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a \in \Z$ be an [[Definition:Integer|integer]].
Let $f: \set {a, a + 1} \to \mathbb A$ be a [[Definition:Real-Valued Function|real-valued function]].
Then the [[Definition:Indexed Summation|indexed... | We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = a}^{a + 1} \map f i
| r = \sum_{i \mathop = a}^a \map f i + \map f {a + 1}
| c = {{Defof|Indexed Summation}}
}}
{{eqn | l =
| r = \map f a + \map f {a + 1}
| c = [[Indexed Summation over Interval of Length One]]
}}
{{end-eqn}}
{{qed}}
[[Cate... | Indexed Summation over Interval of Length Two | https://proofwiki.org/wiki/Indexed_Summation_over_Interval_of_Length_Two | https://proofwiki.org/wiki/Indexed_Summation_over_Interval_of_Length_Two | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Real-Valued Function",
"Definition:Summation/Indexed"
] | [
"Indexed Summation over Interval of Length One",
"Category:Summations"
] |
proofwiki-13646 | Indexed Summation over Interval of Length One | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a \in \Z$ be an integer.
Let $f: \set a \to \mathbb A$ be a mapping on the singleton $\set a$.
Then the indexed summation:
:$\ds \sum_{i \mathop = a}^a \map f i = \map f a$ | We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = a}^a \map f i
| r = \sum_{i \mathop = a}^{a - 1} \map f i + \map f a
| c = {{Defof|Indexed Summation}}
}}
{{eqn | l =
| r = 0 + \map f a
| c = {{Defof|Indexed Summation}}, $a - 1 < a$
}}
{{eqn | l =
| r = \map f a
| c = Identity ... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a \in \Z$ be an [[Definition:Integer|integer]].
Let $f: \set a \to \mathbb A$ be a [[Definition:Mapping|mapping]] on the [[Definition:Singleton|singleton]] $\set a$.
Then the [[Definition:Indexed S... | We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = a}^a \map f i
| r = \sum_{i \mathop = a}^{a - 1} \map f i + \map f a
| c = {{Defof|Indexed Summation}}
}}
{{eqn | l =
| r = 0 + \map f a
| c = {{Defof|Indexed Summation}}, $a - 1 < a$
}}
{{eqn | l =
| r = \map f a
| c = [[Identit... | Indexed Summation over Interval of Length One | https://proofwiki.org/wiki/Indexed_Summation_over_Interval_of_Length_One | https://proofwiki.org/wiki/Indexed_Summation_over_Interval_of_Length_One | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Mapping",
"Definition:Singleton",
"Definition:Summation/Indexed"
] | [
"Identity Element of Addition on Numbers"
] |
proofwiki-13647 | Change of Variables in Indexed Summation | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a, b, c, d$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a mapping.
Let $g: \closedint c d \to \closedint a b$ be a bijection.
Then we have an equality of index... | Because $g : \closedint c d \to \closedint a b$ is a bijection, these sets are equivalent.
By Cardinality of Integer Interval, $\closedint a b$ has cardinality $b - a + 1$.
Thus:
:$b - a + 1 = d - c + 1$
Thus
:$c - a = d - b$
By Indexed Summation over Translated Interval:
:$\ds \sum_{i \mathop = c}^d \map f {\map g i} ... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a, b, c, d$ be [[Definition:Integer|integers]].
Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a [[Def... | Because $g : \closedint c d \to \closedint a b$ is a [[Definition:Bijection|bijection]], these [[Definition:Set|sets]] are [[Definition:Set Equivalence|equivalent]].
By [[Cardinality of Integer Interval]], $\closedint a b$ has [[Definition:Cardinality of Finite Set|cardinality]] $b - a + 1$.
Thus:
:$b - a + 1 = d - c... | Change of Variables in Indexed Summation | https://proofwiki.org/wiki/Change_of_Variables_in_Indexed_Summation | https://proofwiki.org/wiki/Change_of_Variables_in_Indexed_Summation | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Summation/Indexed"
] | [
"Definition:Bijection",
"Definition:Set",
"Definition:Set Equivalence",
"Cardinality of Integer Interval",
"Definition:Cardinality/Finite",
"Indexed Summation over Translated Interval",
"Translation of Integer Interval is Bijection",
"Definition:Mapping",
"Definition:Bijection",
"Composite of Bije... |
proofwiki-13648 | Translation of Integer Interval is Bijection | Let $a, b, c \in \Z$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Then the mapping $T: \closedint a b \to \closedint {a + c} {b + c}$ defined as:
:$\map T k = k + c$
is a bijection. | Note that if $k \in \closedint a b$, then indeed $k + c \in \closedint {a + c} {b + c}$. | Let $a, b, c \in \Z$ be [[Definition:Integer|integers]].
Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Then the [[Definition:Mapping|mapping]] $T: \closedint a b \to \closedint {a + c} {b + c}$ defined as:
:$\map T k = k + c$
is a [[Definition:Bijection|bijecti... | Note that if $k \in \closedint a b$, then indeed $k + c \in \closedint {a + c} {b + c}$. | Translation of Integer Interval is Bijection | https://proofwiki.org/wiki/Translation_of_Integer_Interval_is_Bijection | https://proofwiki.org/wiki/Translation_of_Integer_Interval_is_Bijection | [
"Set Theory"
] | [
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Bijection"
] | [] |
proofwiki-13649 | Indexed Summation without First Term | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a$ and $b$ be integers with $a \le b$.
Let $\closedint a b$ be the integer interval between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a mapping.
Then we have an equality of indexed summations:
:$\ds \sum_{i \mathop = a}^b \map ... | The proof goes by induction on $b$. | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a$ and $b$ be [[Definition:Integer|integers]] with $a \le b$.
Let $\closedint a b$ be the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ ... | The proof goes by [[Principle of Mathematical Induction|induction]] on $b$. | Indexed Summation without First Term | https://proofwiki.org/wiki/Indexed_Summation_without_First_Term | https://proofwiki.org/wiki/Indexed_Summation_without_First_Term | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Summation/Indexed"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-13650 | Summation over Interval equals Indexed Summation | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a, b \in \Z$ be integers.
Let $\closedint a b$ be the integer interval between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a mapping.
Then the summation over the finite set $\closedint a b$ equals the indexed summation from $a$ t... | By Cardinality of Integer Interval, $\closedint a b$ has cardinality $b - a + 1$.
By Translation of Integer Interval is Bijection, the mapping $T : \closedint 0 {b - a} \to \closedint a b$ defined as:
:$\map T k = k + a$
is a bijection.
By definition of summation:
:$\ds \sum_{k \mathop \in \closedint a b} \map f k = \s... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Let $\closedint a b$ be the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a [[Defini... | By [[Cardinality of Integer Interval]], $\closedint a b$ has [[Definition:Cardinality of Finite Set|cardinality]] $b - a + 1$.
By [[Translation of Integer Interval is Bijection]], the [[Definition:Mapping|mapping]] $T : \closedint 0 {b - a} \to \closedint a b$ defined as:
:$\map T k = k + a$
is a [[Definition:Bijectio... | Summation over Interval equals Indexed Summation | https://proofwiki.org/wiki/Summation_over_Interval_equals_Indexed_Summation | https://proofwiki.org/wiki/Summation_over_Interval_equals_Indexed_Summation | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Summation",
"Definition:Finite Set",
"Definition:Summation/Indexed"
] | [
"Cardinality of Integer Interval",
"Definition:Cardinality/Finite",
"Translation of Integer Interval is Bijection",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Summation",
"Indexed Summation over Translated Interval",
"Category:Summations"
] |
proofwiki-13651 | Hardy-Ramanujan Number/Examples/1729 | The $2$nd Hardy-Ramanujan number $\map {\operatorname {Ta}} 2$ is $1729$:
{{begin-eqn}}
{{eqn | l = 1729
| r = 12^3 + 1^3
| c =
}}
{{eqn | r = 10^3 + 9^3
| c =
}}
{{end-eqn}} | We wish to demonstrate that $1729$ is the $2$nd '''Hardy-Ramanujan number''' making it the smallest positive integer which can be expressed as the sum of $2$ positive cubes in $2$ different ways.
To accomplish this, we will need to inspect $66$ sums of $a^3 + b^3$ starting with $1^3 + 1^3$ and ending with $11^3 + 11^3$... | The $2$nd [[Definition:Hardy-Ramanujan Number|Hardy-Ramanujan number]] $\map {\operatorname {Ta}} 2$ is $1729$:
{{begin-eqn}}
{{eqn | l = 1729
| r = 12^3 + 1^3
| c =
}}
{{eqn | r = 10^3 + 9^3
| c =
}}
{{end-eqn}} | We wish to demonstrate that $1729$ is the $2$nd '''[[Definition:Hardy-Ramanujan Number|Hardy-Ramanujan number]]''' making it the smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Positive Integer|positive]] [[Definition:Cube N... | Hardy-Ramanujan Number/Examples/1729 | https://proofwiki.org/wiki/Hardy-Ramanujan_Number/Examples/1729 | https://proofwiki.org/wiki/Hardy-Ramanujan_Number/Examples/1729 | [
"Hardy-Ramanujan Numbers",
"Taxicab Numbers"
] | [
"Definition:Hardy-Ramanujan Number"
] | [
"Definition:Hardy-Ramanujan Number",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Positive/Integer",
"Definition:Cube Number",
"Integer Addition is Commutative",
"Definition:Cube Number",
"Closed Form for Triangular Numbers",
"Definition:Strictly Positive/Integer",
"F... |
proofwiki-13652 | Cardinality of Integer Interval | Let $a, b \in \Z$ be integers.
Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$.
Then $\left[{a \,.\,.\, b}\right]$ is finite and its cardinality equals:
:$\begin{cases}
b - a + 1 & : b \ge a - 1 \\
0 & : b \le a - 1
\end{cases}$ | Let $b < a$.
Then $\left[{a \,.\,.\, b}\right]$ is empty.
By Empty Set is Finite, $\left[{a \,.\,.\, b}\right]$ is finite.
By Cardinality of Empty Set, $\left[{a \,.\,.\, b}\right]$ has cardinality $0$.
Let $b \ge a$.
By Translation of Integer Interval is Bijection, there exists a bijection between $\left[{a \,.\,.\, b... | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Let $\left[{a \,.\,.\, b}\right]$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Then $\left[{a \,.\,.\, b}\right]$ is [[Definition:Finite Set|finite]] and its [[Definition:Cardinality of Finite Set|cardinality]] equals:
:$\begin... | Let $b < a$.
Then $\left[{a \,.\,.\, b}\right]$ is [[Definition:Empty Set|empty]].
By [[Empty Set is Finite]], $\left[{a \,.\,.\, b}\right]$ is [[Definition:Finite Set|finite]].
By [[Cardinality of Empty Set]], $\left[{a \,.\,.\, b}\right]$ has [[Definition:Cardinality of Finite Set|cardinality]] $0$.
Let $b \ge a$... | Cardinality of Integer Interval | https://proofwiki.org/wiki/Cardinality_of_Integer_Interval | https://proofwiki.org/wiki/Cardinality_of_Integer_Interval | [
"Set Theory"
] | [
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Finite Set",
"Definition:Cardinality/Finite"
] | [
"Definition:Empty Set",
"Empty Set is Finite",
"Definition:Finite Set",
"Cardinality of Empty Set",
"Definition:Cardinality/Finite",
"Translation of Integer Interval is Bijection",
"Definition:Bijection",
"Definition:Finite Set",
"Definition:Cardinality/Finite",
"Category:Set Theory"
] |
proofwiki-13653 | Indexed Summation of Sum of Mappings | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a, b$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Let $f, g: \closedint a b \to \mathbb A$ be mappings.
Let $h = f + g$ be their pointwise sum.
Then we have the equality of indexed summations:
:$\ds \... | The proof proceeds by induction on $b$.
For all $b \in \Z_{\ge 0}$, let $\map P b$ be the proposition:
:$\ds \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$ | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a, b$ be [[Definition:Integer|integers]].
Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f, g: \closedint a b \to \mathbb A$ be [[Definiti... | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $b$.
For all $b \in \Z_{\ge 0}$, let $\map P b$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$ | Indexed Summation of Sum of Mappings | https://proofwiki.org/wiki/Indexed_Summation_of_Sum_of_Mappings | https://proofwiki.org/wiki/Indexed_Summation_of_Sum_of_Mappings | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Pointwise Addition",
"Definition:Summation/Indexed"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-13654 | Summation of Sum of Mappings on Finite Set | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f, g: S \to \mathbb A$ be mappings.
Let $h = f + g$ be their sum.
Then we have the equality of summations on finite sets:
:$\ds \sum_{s \mathop \in S} \map h s = \sum_{s \mathop \in S} \map f s + \sum_{s \mathop \i... | Let $n$ be the cardinality of $S$.
Let $\sigma: \N_{< n} \to S$ be a bijection, where $\N_{< n}$ is an initial segment of the natural numbers.
By definition of summation, we have to prove the following equality of indexed summations:
:$\ds \sum_{i \mathop = 0}^{n - 1} \map h {\map \sigma i} = \sum_{i \mathop = 0}^{n - ... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f, g: S \to \mathbb A$ be [[Definition:Mapping|mappings]].
Let $h = f + g$ be their [[Definition:Sum of Mappings|sum]].
Then we have the equality... | Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$.
Let $\sigma: \N_{< n} \to S$ be a [[Definition:Bijection|bijection]], where $\N_{< n}$ is an [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]].
By definition of [[Definition:Summation|summation]], we hav... | Summation of Sum of Mappings on Finite Set | https://proofwiki.org/wiki/Summation_of_Sum_of_Mappings_on_Finite_Set | https://proofwiki.org/wiki/Summation_of_Sum_of_Mappings_on_Finite_Set | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Sum of Mappings",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Definition:Cardinality/Finite",
"Definition:Bijection",
"Definition:Initial Segment of Natural Numbers",
"Definition:Summation",
"Definition:Summation/Indexed",
"Sum of Mappings Composed with Mapping",
"Indexed Summation of Sum of Mappings"
] |
proofwiki-13655 | Indexed Summation of Multiple of Mapping | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a, b$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a mapping.
Let $\lambda \in \mathbb A$.
Let $g = \lambda \cdot f$ be the product of $f$ with $\lambda$.
Then ... | The proof goes by induction on $b$. | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a, b$ be [[Definition:Integer|integers]].
Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f: \closedint a b \to \mathbb A$ be a [[Definitio... | The proof goes by [[Principle of Mathematical Induction|induction]] on $b$. | Indexed Summation of Multiple of Mapping | https://proofwiki.org/wiki/Indexed_Summation_of_Multiple_of_Mapping | https://proofwiki.org/wiki/Indexed_Summation_of_Multiple_of_Mapping | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Product of Mapping with Scalar",
"Definition:Summation/Indexed"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-13656 | Summation of Multiple of Mapping on Finite Set | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f: S \to \mathbb A$ be a mapping.
Let $\lambda \in \mathbb A$.
Let $g = \lambda \cdot f$ be the product of $f$ with $\lambda$.
Then we have the equality of summations on finite sets:
:$\ds \sum_{s \mathop \in S} \m... | Let $n$ be the cardinality of $S$.
Let $\sigma: \N_{< n} \to S$ be a bijection, where $\N_{< n}$ is an initial segment of the natural numbers.
By definition of summation, we have to prove the following equality of indexed summations:
:$\ds \sum_{i \mathop = 0}^{n - 1} \map g {\map \sigma i} = \lambda \cdot \sum_{i \mat... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Let $\lambda \in \mathbb A$.
Let $g = \lambda \cdot f$ be the [[Definition:Product of Map... | Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$.
Let $\sigma: \N_{< n} \to S$ be a [[Definition:Bijection|bijection]], where $\N_{< n}$ is an [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]].
By definition of [[Definition:Summation|summation]], we hav... | Summation of Multiple of Mapping on Finite Set | https://proofwiki.org/wiki/Summation_of_Multiple_of_Mapping_on_Finite_Set | https://proofwiki.org/wiki/Summation_of_Multiple_of_Mapping_on_Finite_Set | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Product of Mapping with Scalar",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Definition:Cardinality/Finite",
"Definition:Bijection",
"Definition:Initial Segment of Natural Numbers",
"Definition:Summation",
"Definition:Summation/Indexed",
"Multiple of Mapping Composed with Mapping",
"Indexed Summation of Multiple of Mapping"
] |
proofwiki-13657 | Linear Combination of Indexed Summations | Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$.
Let $a,b$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Let $f, g : \closedint a b \to \mathbb A$ be mappings.
Let $\lambda, \mu \in \mathbb A$.
Let $\lambda \cdot f + \mu \cdot g$ be the sum of the product of $... | We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = a}^b \paren {\lambda \cdot \map f i + \mu \cdot \map g i}
| r = \sum_{i \mathop = a}^b \paren {\lambda \cdot \map f i} + \sum_{i \mathop = a}^b \paren {\mu \cdot \map g i}
| c = Indexed Summation of Sum of Mappings
}}
{{eqn | r = \lambda \cdot \sum_{i \ma... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N,\Z,\Q,\R,\C$.
Let $a,b$ be [[Definition:Integer|integers]].
Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f, g : \closedint a b \to \mathbb A$ be [[Definition:M... | We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = a}^b \paren {\lambda \cdot \map f i + \mu \cdot \map g i}
| r = \sum_{i \mathop = a}^b \paren {\lambda \cdot \map f i} + \sum_{i \mathop = a}^b \paren {\mu \cdot \map g i}
| c = [[Indexed Summation of Sum of Mappings]]
}}
{{eqn | r = \lambda \cdot \sum_{i... | Linear Combination of Indexed Summations | https://proofwiki.org/wiki/Linear_Combination_of_Indexed_Summations | https://proofwiki.org/wiki/Linear_Combination_of_Indexed_Summations | [
"Summations"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Sum of Mappings",
"Definition:Product of Mapping with Scalar",
"Definition:Product of Mapping with Scalar",
"Definition:Summation/Indexed"
] | [
"Indexed Summation of Sum of Mappings",
"Indexed Summation of Multiple of Mapping"
] |
proofwiki-13658 | Triangle Inequality for Indexed Summations | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a,b$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Let $f : \closedint a b \to \mathbb A$ be a mapping.
Let $\size {\, \cdot \,}$ denote the standard absolute value.
Let $\vert f \vert$ be the absolute ... | The proof goes by induction on $b$. | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $a,b$ be [[Definition:Integer|integers]].
Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f : \closedint a b \to \mathbb A$ be a [[Definitio... | The proof goes by [[Principle of Mathematical Induction|induction]] on $b$. | Triangle Inequality for Indexed Summations | https://proofwiki.org/wiki/Triangle_Inequality_for_Indexed_Summations | https://proofwiki.org/wiki/Triangle_Inequality_for_Indexed_Summations | [
"Summations",
"Triangle Inequality"
] | [
"Definition:Number",
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Mapping",
"Definition:Standard Absolute Value",
"Definition:Absolute Value of Mapping",
"Definition:Inequality",
"Definition:Summation/Indexed"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-13659 | Triangle Inequality for Summation over Finite Set | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f : S \to \mathbb A$ be a mapping.
Let $\size {\, \cdot\,}$ denote the standard absolute value.
Let $\size f$ be the absoute value of $f$.
Then we have the inequality of summations on finite sets:
:$\ds \size {\sum... | Let $n$ be the cardinality of $S$.
Let $\sigma: \N_{< n} \to S$ be a bijection, where $\N_{<n}$ is an initial segment of the natural numbers.
By definition of summation, we have to prove the following inequality of indexed summations:
:$\ds \size {\sum_{i \mathop = 0}^{n - 1} \map f {\map \sigma i} } \le \sum_{i \matho... | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f : S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Let $\size {\, \cdot\,}$ denote the [[Definition:Standard Absolute Value|standard absolut... | Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $S$.
Let $\sigma: \N_{< n} \to S$ be a [[Definition:Bijection|bijection]], where $\N_{<n}$ is an [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]].
By definition of [[Definition:Summation|summation]], we have... | Triangle Inequality for Summation over Finite Set | https://proofwiki.org/wiki/Triangle_Inequality_for_Summation_over_Finite_Set | https://proofwiki.org/wiki/Triangle_Inequality_for_Summation_over_Finite_Set | [
"Summations",
"Triangle Inequality"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Standard Absolute Value",
"Definition:Absolute Value of Mapping",
"Definition:Inequality",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Definition:Cardinality/Finite",
"Definition:Bijection",
"Definition:Initial Segment of Natural Numbers",
"Definition:Summation",
"Definition:Inequality",
"Definition:Summation/Indexed",
"Absolute Value of Mapping Composed with Mapping",
"Triangle Inequality for Indexed Summations"
] |
proofwiki-13660 | Exchange of Order of Indexed Summations/Rectangular Domain | Let $D = \closedint a b \times \closedint c d$ be the cartesian product.
Let $f: D \to \mathbb A$ be a mapping
Then we have an equality of indexed summations:
:$\ds \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d \map f {i, j} = \sum_{j \mathop = c}^d \sum_{i \mathop = a}^b \map f {i, j}$ | The proof proceeds by induction on $d$. | Let $D = \closedint a b \times \closedint c d$ be the [[Definition:Cartesian Product|cartesian product]].
Let $f: D \to \mathbb A$ be a [[Definition:Mapping|mapping]]
Then we have an equality of [[Definition:Indexed Summation|indexed summations]]:
:$\ds \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d \map f {i, j} = \... | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $d$. | Exchange of Order of Indexed Summations/Rectangular Domain | https://proofwiki.org/wiki/Exchange_of_Order_of_Indexed_Summations/Rectangular_Domain | https://proofwiki.org/wiki/Exchange_of_Order_of_Indexed_Summations/Rectangular_Domain | [
"Summations"
] | [
"Definition:Cartesian Product",
"Definition:Mapping",
"Definition:Summation/Indexed"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-13661 | Exchange of Order of Summations over Finite Sets/Cartesian Product | Let $f: S \times T \to \mathbb A$ be a mapping.
Then we have an equality of summations over finite sets:
:$\ds \sum_{s \mathop \in S} \sum_{t \mathop \in T} \map f {s, t} = \sum_{t \mathop \in T} \sum_{s \mathop \in S} \map f {s, t}$ | Let $n$ be the cardinality of $T$.
The proof goes by induction on $n$.
=== Basis for the Induction ===
Let $n = 0$.
{{finish}}
=== Induction Step ===
Let $n > 0$.
Let $t \in T$.
Use Cardinality of Set minus Singleton
{{ProofWanted}} | Let $f: S \times T \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Then we have an equality of [[Definition:Summation|summations]] over [[Definition:Finite Set|finite sets]]:
:$\ds \sum_{s \mathop \in S} \sum_{t \mathop \in T} \map f {s, t} = \sum_{t \mathop \in T} \sum_{s \mathop \in S} \map f {s, t}$ | Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $T$.
The proof goes by [[Principle of Mathematical Induction|induction]] on $n$.
=== Basis for the Induction ===
Let $n = 0$.
{{finish}}
=== Induction Step ===
Let $n > 0$.
Let $t \in T$.
Use [[Cardinality of Set minus Singleton]]
{{Proof... | Exchange of Order of Summations over Finite Sets/Cartesian Product/Proof 3 | https://proofwiki.org/wiki/Exchange_of_Order_of_Summations_over_Finite_Sets/Cartesian_Product | https://proofwiki.org/wiki/Exchange_of_Order_of_Summations_over_Finite_Sets/Cartesian_Product/Proof_3 | [
"Summations",
"Exchange of Order of Summations over Finite Sets"
] | [
"Definition:Mapping",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Definition:Cardinality/Finite",
"Principle of Mathematical Induction",
"Cardinality of Set minus Singleton"
] |
proofwiki-13662 | Exchange of Order of Summations over Finite Sets | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S, T$ be finite sets.
Let $S \times T$ be their cartesian product. | Let $n$ be the cardinality of $T$.
The proof goes by induction on $n$.
=== Basis for the Induction ===
Let $n = 0$.
{{finish}}
=== Induction Step ===
Let $n > 0$.
Let $t \in T$.
Use Cardinality of Set minus Singleton
{{ProofWanted}} | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S, T$ be [[Definition:Finite Set|finite sets]].
Let $S \times T$ be their [[Definition:Cartesian Product|cartesian product]]. | Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $T$.
The proof goes by [[Principle of Mathematical Induction|induction]] on $n$.
=== Basis for the Induction ===
Let $n = 0$.
{{finish}}
=== Induction Step ===
Let $n > 0$.
Let $t \in T$.
Use [[Cardinality of Set minus Singleton]]
{{Proof... | Exchange of Order of Summations over Finite Sets/Cartesian Product/Proof 3 | https://proofwiki.org/wiki/Exchange_of_Order_of_Summations_over_Finite_Sets | https://proofwiki.org/wiki/Exchange_of_Order_of_Summations_over_Finite_Sets/Cartesian_Product/Proof_3 | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Cartesian Product"
] | [
"Definition:Cardinality/Finite",
"Principle of Mathematical Induction",
"Cardinality of Set minus Singleton"
] |
proofwiki-13663 | Sum over Complement of Finite Set | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f: S \to \mathbb A$ be a mapping.
Let $T \subseteq S$ be a subset.
Let $S \setminus T$ be its relative complement.
Then we have the equality of summations over finite sets:
:$\ds \sum_{s \mathop \in S \setminus T} ... | Note that by Subset of Finite Set is Finite, $T$ is indeed finite.
By Set is Disjoint Union of Subset and Relative Complement, $S$ is the disjoint union of $S \setminus T$ and $T$.
The result now follows from Sum over Disjoint Union of Finite Sets.
{{qed}} | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]].
Let $S \setminus T$ be its [[Defin... | Note that by [[Subset of Finite Set is Finite]], $T$ is indeed [[Definition:Finite Set|finite]].
By [[Set is Disjoint Union of Subset and Relative Complement]], $S$ is the [[Definition:Disjoint Union|disjoint union]] of $S \setminus T$ and $T$.
The result now follows from [[Sum over Disjoint Union of Finite Sets]].
{... | Sum over Complement of Finite Set | https://proofwiki.org/wiki/Sum_over_Complement_of_Finite_Set | https://proofwiki.org/wiki/Sum_over_Complement_of_Finite_Set | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Subset",
"Definition:Relative Complement",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"Set is Disjoint Union of Subset and Relative Complement",
"Definition:Disjoint Union",
"Sum over Disjoint Union of Finite Sets"
] |
proofwiki-13664 | Mapping Defines Additive Function of Subalgebra of Power Set | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f: S \to \mathbb A$ be a mapping.
Let $B$ be an algebra of sets over $S$.
Define $\Sigma: B \to \mathbb A$ using summation as:
:$\ds \map \Sigma T = \sum_{t \mathop \in T} \map f t$
for $T\subseteq S$.
Then $\Sigma... | Note that by Subset of Finite Set is Finite, $B$ consists of finite sets.
The result now follows from Sum over Disjoint Union of Finite Sets.
{{qed}} | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Let $B$ be an [[Definition:Algebra of Sets|algebra of sets]] over $S$.
Define $\Sigma: B ... | Note that by [[Subset of Finite Set is Finite]], $B$ consists of [[Definition:Finite Set|finite sets]].
The result now follows from [[Sum over Disjoint Union of Finite Sets]].
{{qed}} | Mapping Defines Additive Function of Subalgebra of Power Set | https://proofwiki.org/wiki/Mapping_Defines_Additive_Function_of_Subalgebra_of_Power_Set | https://proofwiki.org/wiki/Mapping_Defines_Additive_Function_of_Subalgebra_of_Power_Set | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Algebra of Sets",
"Definition:Summation",
"Definition:Additive Function (Measure Theory)"
] | [
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"Sum over Disjoint Union of Finite Sets"
] |
proofwiki-13665 | Sum over Union of Finite Sets | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ and $T$ be finite sets.
Let $f: S \cup T \to \mathbb A$ be a mapping.
Then we have the equality of summations over finite sets:
:$\ds \sum_{u \mathop \in S \mathop \cup T} \map f u = \sum_{s \mathop \in S} \map f s + \sum_{t \mathop \in... | Follows from:
:Mapping Defines Additive Function of Subalgebra of Power Set
:Power Set is Algebra of Sets
:Inclusion-Exclusion Principle
{{qed}} | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ and $T$ be [[Definition:Finite Set|finite sets]].
Let $f: S \cup T \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Then we have the equality of [[Definition:Summation|summations]] over [[Def... | Follows from:
:[[Mapping Defines Additive Function of Subalgebra of Power Set]]
:[[Power Set is Algebra of Sets]]
:[[Inclusion-Exclusion Principle]]
{{qed}} | Sum over Union of Finite Sets | https://proofwiki.org/wiki/Sum_over_Union_of_Finite_Sets | https://proofwiki.org/wiki/Sum_over_Union_of_Finite_Sets | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Mapping Defines Additive Function of Subalgebra of Power Set",
"Power Set is Algebra of Sets",
"Inclusion-Exclusion Principle"
] |
proofwiki-13666 | Summation over Finite Set Equals Summation over Support | Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f: S \to \mathbb A$ be a mapping.
Let $\map \supp f$ be its support.
Then we have an equality of summations over finite sets:
:$\ds \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in \map \supp f} \map f s$ | Note that by Subset of Finite Set is Finite, $\map \supp f$ is indeed finite.
The result now follows from:
* Sum over Complement of Finite Set
* Sum of Zero over Finite Set
* Identity Element of Addition on Numbers
{{qed}} | Let $\mathbb A$ be one of the [[Definition:Standard Number System|standard number systems]] $\N, \Z, \Q, \R, \C$.
Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to \mathbb A$ be a [[Definition:Mapping|mapping]].
Let $\map \supp f$ be its [[Definition:Support of Mapping to Algebraic Structure|support]]... | Note that by [[Subset of Finite Set is Finite]], $\map \supp f$ is indeed [[Definition:Finite Set|finite]].
The result now follows from:
* [[Sum over Complement of Finite Set]]
* [[Sum of Zero over Finite Set]]
* [[Identity Element of Addition on Numbers]]
{{qed}} | Summation over Finite Set Equals Summation over Support | https://proofwiki.org/wiki/Summation_over_Finite_Set_Equals_Summation_over_Support | https://proofwiki.org/wiki/Summation_over_Finite_Set_Equals_Summation_over_Support | [
"Summations"
] | [
"Definition:Number",
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Support of Mapping to Algebraic Structure",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"Sum over Complement of Finite Set",
"Summation of Zero/Finite Set",
"Identity Element of Addition on Numbers"
] |
proofwiki-13667 | Summation of Zero/Indexed Summation | Let $a, b$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Let $f_0 : \closedint a b \to \mathbb A$ be the zero mapping.
Then the indexed summation of $0$ from $a$ to $b$ equals zero:
:$\ds \sum_{i \mathop = a}^b \map {f_0} i = 0$ | At least three proofs are possible:
* by induction, using Identity Element of Addition on Numbers
* using Indexed Summation of Multiple of Mapping
* using Indexed Summation of Sum of Mappings
{{ProofWanted}}
Category:Summations
drp3mu7oymvpogx6al6gdb0lpczczok | Let $a, b$ be [[Definition:Integer|integers]].
Let $\closedint a b$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $f_0 : \closedint a b \to \mathbb A$ be the [[Definition:Zero Mapping|zero mapping]].
Then the [[Definition:Indexed Summation|indexed summation]] of $0$ from $a$ t... | At least three proofs are possible:
* by induction, using [[Identity Element of Addition on Numbers]]
* using [[Indexed Summation of Multiple of Mapping]]
* using [[Indexed Summation of Sum of Mappings]]
{{ProofWanted}}
[[Category:Summations]]
drp3mu7oymvpogx6al6gdb0lpczczok | Summation of Zero/Indexed Summation | https://proofwiki.org/wiki/Summation_of_Zero/Indexed_Summation | https://proofwiki.org/wiki/Summation_of_Zero/Indexed_Summation | [
"Summations"
] | [
"Definition:Integer",
"Definition:Closed Interval/Integer Interval",
"Definition:Zero Mapping",
"Definition:Summation/Indexed",
"Definition:Zero (Number)"
] | [
"Identity Element of Addition on Numbers",
"Indexed Summation of Multiple of Mapping",
"Indexed Summation of Sum of Mappings",
"Category:Summations"
] |
proofwiki-13668 | Summation of Zero/Finite Set | Let $S$ be a finite set.
Let $0 : S \to \mathbb A$ be the zero mapping.
{{explain|Presumably the above is a constant mapping on $0$ -- needs to be made explicit.}}
Then the summation of $0$ over $S$ equals zero:
:$\ds \sum_{s \mathop \in S} 0 \left({s}\right) = 0$ | At least three proofs are possible:
:using the definition of summation and Indexed Summation of Zero
:using Indexed Summation of Sum of Mappings
:using Summation of Multiple of Mapping on Finite Set.
{{ProofWanted}}
Category:Summations
h5knszknmzvjg820yqugvmdcw8um0sv | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $0 : S \to \mathbb A$ be the [[Definition:Zero Mapping|zero mapping]].
{{explain|Presumably the above is a [[Definition:Constant Mapping|constant mapping]] on $0$ -- needs to be made explicit.}}
Then the [[Definition:Summation|summation]] of $0$ over $S$ equals... | At least three proofs are possible:
:using the definition of [[Definition:Summation|summation]] and [[Indexed Summation of Zero]]
:using [[Indexed Summation of Sum of Mappings]]
:using [[Summation of Multiple of Mapping on Finite Set]].
{{ProofWanted}}
[[Category:Summations]]
h5knszknmzvjg820yqugvmdcw8um0sv | Summation of Zero/Finite Set | https://proofwiki.org/wiki/Summation_of_Zero/Finite_Set | https://proofwiki.org/wiki/Summation_of_Zero/Finite_Set | [
"Summations"
] | [
"Definition:Finite Set",
"Definition:Zero Mapping",
"Definition:Constant Mapping",
"Definition:Summation",
"Definition:Zero (Number)"
] | [
"Definition:Summation",
"Summation of Zero/Indexed Summation",
"Indexed Summation of Sum of Mappings",
"Summation of Multiple of Mapping on Finite Set",
"Category:Summations"
] |
proofwiki-13669 | Summation of Zero/Set | Let $S$ be a set.
Let $0: S \to \mathbb A$ be the zero mapping.
Then the summation with finite support of $0$ over $S$ equals zero:
:$\ds \sum_{s \mathop \in S} \map 0 s = 0$ | By Support of Zero Mapping, the support of $0$ is empty.
By Empty Set is Finite, the support of $0$ is indeed finite.
By Summation over Empty Set:
:$\ds \sum_{s \mathop \in S} \map 0 s = \sum_{s \mathop \in \O} \map 0 s = 0$
{{qed}}
Category:Summations
7euo1h4t8r8qpdusxsnkr2bmrj3ezg6 | Let $S$ be a [[Definition:Set|set]].
Let $0: S \to \mathbb A$ be the [[Definition:Zero Mapping|zero mapping]].
Then the [[Definition:Summation over Set with Finite Support|summation with finite support]] of $0$ over $S$ equals [[Definition:Zero of Standard Number System|zero]]:
:$\ds \sum_{s \mathop \in S} \map 0 s ... | By [[Support of Zero Mapping]], the [[Definition:Support of Mapping to Algebraic Structure|support]] of $0$ is [[Definition:Empty Set|empty]].
By [[Empty Set is Finite]], the [[Definition:Support of Mapping to Algebraic Structure|support]] of $0$ is indeed [[Definition:Finite Set|finite]].
By [[Summation over Empty S... | Summation of Zero/Set | https://proofwiki.org/wiki/Summation_of_Zero/Set | https://proofwiki.org/wiki/Summation_of_Zero/Set | [
"Summations"
] | [
"Definition:Set",
"Definition:Zero Mapping",
"Definition:Summation/Finite Support",
"Definition:Zero (Number)"
] | [
"Support of Zero Mapping",
"Definition:Support of Mapping to Algebraic Structure",
"Definition:Empty Set",
"Empty Set is Finite",
"Definition:Support of Mapping to Algebraic Structure",
"Definition:Finite Set",
"Summation over Empty Set",
"Category:Summations"
] |
proofwiki-13670 | Exchange of Order of Summations over Finite Sets/Subset of Cartesian Product | Let $D\subset S \times T$ be a subset.
Let $\pi_1 : D \to S$ and $\pi_2 : D \to T$ be the restrictions of the projections of $S\times T$.
Then we have an equality of summations over finite sets:
:$\ds \sum_{s \mathop \in S} \sum_{t \mathop \in \map {\pi_2} {\map {\pi_1^{-1} } s} } \map f {s, t} = \sum_{t \mathop \in T}... | Define an extension $\overline f$ of $f$ to $S \times T$ by:
:$\map {\overline f} {s, t} = \begin{cases}
\map f {s, t} & : \tuple {s, t} \in D \\
0 & : \tuple {s, t} \notin D
\end{cases}$
Then for all $s \in S$, by:
:Preimage of Disjoint Union is Disjoint Union
:Sum over Disjoint Union of Finite Sets
:Summation over Fi... | Let $D\subset S \times T$ be a [[Definition:Subset|subset]].
Let $\pi_1 : D \to S$ and $\pi_2 : D \to T$ be the [[Definition:Restriction of Mapping|restrictions]] of the [[Definition:Projection from Cartesian Product|projections]] of $S\times T$.
Then we have an equality of [[Definition:Summation|summations]] over [... | Define an [[Definition:Extension of Mapping|extension]] $\overline f$ of $f$ to $S \times T$ by:
:$\map {\overline f} {s, t} = \begin{cases}
\map f {s, t} & : \tuple {s, t} \in D \\
0 & : \tuple {s, t} \notin D
\end{cases}$
Then for all $s \in S$, by:
:[[Preimage of Disjoint Union is Disjoint Union]]
:[[Sum over Disj... | Exchange of Order of Summations over Finite Sets/Subset of Cartesian Product | https://proofwiki.org/wiki/Exchange_of_Order_of_Summations_over_Finite_Sets/Subset_of_Cartesian_Product | https://proofwiki.org/wiki/Exchange_of_Order_of_Summations_over_Finite_Sets/Subset_of_Cartesian_Product | [
"Summations"
] | [
"Definition:Subset",
"Definition:Restriction/Mapping",
"Definition:Projection from Cartesian Product",
"Definition:Summation",
"Definition:Finite Set"
] | [
"Definition:Extension of Mapping",
"Preimage of Disjoint Union is Disjoint Union",
"Sum over Disjoint Union of Finite Sets",
"Summation of Zero/Finite Set",
"Exchange of Order of Summations over Finite Sets/Cartesian Product",
"Category:Summations"
] |
proofwiki-13671 | Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary | Let $R$ be a commutative ring.
Let $\struct {S, +, *}$ be a ring with unity.
Let $f: R \to S$ be a ring homomorphism.
Let $\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$.
Then $\struct {S_R, *}$ is a unitary algebra. | By definition, the multiplication of $\struct {S_R, *}$ is the ring product of $S$.
Thus it follows immediately from the fact that $S$ is a ring with unity, that $\struct {S_R, *}$ is a unitary algebra.
{{qed}}
Category:Algebras
Category:Unital Algebras
3ecweu0z3ty2ohlojp54ic7qg4wpq4n | Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {S, +, *}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $f: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let $\struct {S_R, *}$ be the [[Definition:Algebra Defined by Ring Homomorphism|algebra defined by the ... | By definition, the [[Definition:Multiplication of Algebra|multiplication]] of $\struct {S_R, *}$ is the [[Definition:Ring Product|ring product]] of $S$.
Thus it follows immediately from the fact that $S$ is a [[Definition:Ring with Unity|ring with unity]], that $\struct {S_R, *}$ is a [[Definition:Unitary Algebra|unit... | Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary | https://proofwiki.org/wiki/Algebra_Defined_by_Ring_Homomorphism_on_Ring_with_Unity_is_Unitary | https://proofwiki.org/wiki/Algebra_Defined_by_Ring_Homomorphism_on_Ring_with_Unity_is_Unitary | [
"Algebras",
"Unital Algebras"
] | [
"Definition:Commutative Ring",
"Definition:Ring with Unity",
"Definition:Ring Homomorphism",
"Definition:Algebra Defined by Ring Homomorphism",
"Definition:Unital Algebra"
] | [
"Definition:Multiplication of Algebra",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Ring with Unity",
"Definition:Unital Algebra",
"Category:Algebras",
"Category:Unital Algebras"
] |
proofwiki-13672 | Algebra Defined by Ring Homomorphism is Associative | Let $R$ be a commutative ring.
Let $\struct {S, +, *}$ be a ring with unity.
Let $f: R \to S$ be a ring homomorphism.
Let the image of $f$ be a subset of the center of $S$.
Let $\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$.
Then $\struct {S_R, *}$ is an associative algebra. | By definition, the multiplication of $\struct {S_R, *}$ is the ring product of $S$.
Thus it follows immediately from the fact that $S$ is a ring, that $\struct {S_R, *}$ is an associative algebra.
{{qed}} | Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {S, +, *}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $f: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let the [[Definition:Image of Set under Mapping|image]] of $f$ be a [[Definition:Subset|subset]] of the... | By definition, the [[Definition:Multiplication of Algebra|multiplication]] of $\struct {S_R, *}$ is the [[Definition:Ring Product|ring product]] of $S$.
Thus it follows immediately from the fact that $S$ is a [[Definition:Ring (Abstract Algebra)|ring]], that $\struct {S_R, *}$ is an [[Definition:Associative Algebra|as... | Algebra Defined by Ring Homomorphism is Associative | https://proofwiki.org/wiki/Algebra_Defined_by_Ring_Homomorphism_is_Associative | https://proofwiki.org/wiki/Algebra_Defined_by_Ring_Homomorphism_is_Associative | [
"Algebras",
"Associative Algebras"
] | [
"Definition:Commutative Ring",
"Definition:Ring with Unity",
"Definition:Ring Homomorphism",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subset",
"Definition:Center (Abstract Algebra)/Ring",
"Definition:Algebra Defined by Ring Homomorphism",
"Definition:Associative Algebra"
] | [
"Definition:Multiplication of Algebra",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Ring (Abstract Algebra)",
"Definition:Associative Algebra"
] |
proofwiki-13673 | Algebra Defined by Ring Homomorphism on Commutative Ring is Commutative | Let $R$ be a commutative ring.
Let $\struct {S, +, *}$ be a commutative ring.
Let $f: R \to S$ be a ring homomorphism.
Let $\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$.
Then $\struct {S_R, *}$ is a commutative algebra. | Note that by Center of Commutative Ring, the image of $f$ is indeed a subset of the center of $S$.
By definition, the multiplication of $\struct {S_R, *}$ is the ring product of $S$.
Thus it follows immediately from the fact that $S$ is a ring, that $\struct {S_R, *}$ is a commutative algebra.
{{qed}} | Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {S, +, *}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $f: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let $\struct {S_R, *}$ be the [[Definition:Algebra Defined by Ring Homomorphism|algebra defined by th... | Note that by [[Center of Commutative Ring]], the [[Definition:Image of Set under Mapping|image]] of $f$ is indeed a [[Definition:Subset|subset]] of the [[Definition:Center of Ring|center]] of $S$.
By definition, the [[Definition:Multiplication of Algebra|multiplication]] of $\struct {S_R, *}$ is the [[Definition:Ring ... | Algebra Defined by Ring Homomorphism on Commutative Ring is Commutative | https://proofwiki.org/wiki/Algebra_Defined_by_Ring_Homomorphism_on_Commutative_Ring_is_Commutative | https://proofwiki.org/wiki/Algebra_Defined_by_Ring_Homomorphism_on_Commutative_Ring_is_Commutative | [
"Algebras",
"Commutative Algebras"
] | [
"Definition:Commutative Ring",
"Definition:Commutative Ring",
"Definition:Ring Homomorphism",
"Definition:Algebra Defined by Ring Homomorphism",
"Definition:Commutative Algebra (Abstract Algebra)"
] | [
"Center of Commutative Ring",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subset",
"Definition:Center (Abstract Algebra)/Ring",
"Definition:Multiplication of Algebra",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Ring (Abstract Algebra)",
"Definition:Commutative Algebra (A... |
proofwiki-13674 | Smallest Integer which is Sum of 3 Sixth Powers in 2 Ways | The smallest positive integer which can be expressed as the sum of $3$ sixth powers in $2$ different ways is:
{{begin-eqn}}
{{eqn | l = 160 \, 426 \, 514
| r = 3^6 + 19^6 + 22^6
| c =
}}
{{eqn | r = 10^6 + 15^6 + 23^6
| c =
}}
{{end-eqn}}
Also note that:
{{begin-eqn}}
{{eqn | l = 854
| r = 3^2... | We have that:
{{begin-eqn}}
{{eqn | l = 160 \, 426 \, 514
| r = 729 + 47 \, 045 \, 881 + 113 \, 379 \, 904
| c =
}}
{{eqn | r = 3^6 + 19^6 + 22^6
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 160 \, 426 \, 514
| r = 1 \, 000 \, 000 + 11 \, 390 \, 625 + 148 \, 035 \, 889
| c =
}}
{{eqn ... | The smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Sixth Power|sixth powers]] in $2$ different ways is:
{{begin-eqn}}
{{eqn | l = 160 \, 426 \, 514
| r = 3^6 + 19^6 + 22^6
| c =
}}
{{eqn | r = 10^6 + 15^6 + 23... | We have that:
{{begin-eqn}}
{{eqn | l = 160 \, 426 \, 514
| r = 729 + 47 \, 045 \, 881 + 113 \, 379 \, 904
| c =
}}
{{eqn | r = 3^6 + 19^6 + 22^6
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 160 \, 426 \, 514
| r = 1 \, 000 \, 000 + 11 \, 390 \, 625 + 148 \, 035 \, 889
| c =
}}
{{e... | Smallest Integer which is Sum of 3 Sixth Powers in 2 Ways | https://proofwiki.org/wiki/Smallest_Integer_which_is_Sum_of_3_Sixth_Powers_in_2_Ways | https://proofwiki.org/wiki/Smallest_Integer_which_is_Sum_of_3_Sixth_Powers_in_2_Ways | [
"Sixth Powers",
"160,426,514"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Sixth Power"
] | [] |
proofwiki-13675 | Infinite Number of Integers which are Sum of 3 Sixth Powers in 2 Ways | There exist an infinite number of positive integers which can be expressed as the sum of $3$ sixth powers in $2$ different ways. | There are many parametric solutions to $x^6 + y^6 + z^6 = u^6 + v^6 + w^6$. One is given by:
{{begin-eqn}}
{{eqn | l = x
| r = 2 m^4 + 4 m^3 n - 5 m^2 n^2 - 12 m n^3 - 9 n^4
}}
{{eqn | l = y
| r = 3 m^4 + 9 m^3 n + 18 m^2 n^2 + 21 m n^3 + 9 n^4
}}
{{eqn | l = z
| r = -m^4 - 10 m^3 n - 17 m^2 n^2 - 12 ... | There exist an [[Definition:Infinite Set|infinite number]] of [[Definition:Positive Integer|positive integers]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Sixth Power|sixth powers]] in $2$ different ways. | There are many parametric solutions to $x^6 + y^6 + z^6 = u^6 + v^6 + w^6$. One is given by:
{{begin-eqn}}
{{eqn | l = x
| r = 2 m^4 + 4 m^3 n - 5 m^2 n^2 - 12 m n^3 - 9 n^4
}}
{{eqn | l = y
| r = 3 m^4 + 9 m^3 n + 18 m^2 n^2 + 21 m n^3 + 9 n^4
}}
{{eqn | l = z
| r = -m^4 - 10 m^3 n - 17 m^2 n^2 - 12... | Infinite Number of Integers which are Sum of 3 Sixth Powers in 2 Ways | https://proofwiki.org/wiki/Infinite_Number_of_Integers_which_are_Sum_of_3_Sixth_Powers_in_2_Ways | https://proofwiki.org/wiki/Infinite_Number_of_Integers_which_are_Sum_of_3_Sixth_Powers_in_2_Ways | [
"Sixth Powers",
"160,426,514"
] | [
"Definition:Infinite Set",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Sixth Power"
] | [] |
proofwiki-13676 | Consecutive Primes of form 4n+1 | The sequence of $16$ consecutive prime numbers beginning from $207 \, 622 \, 273$ are all of the form $4 n + 1$. | {{begin-eqn}}
{{eqn | n = 1
| l = 207 \, 622 \, 273
| r = 4 \times 51 \, 905 \, 568 + 1
| c = and is the $11 \, 477 \, 482$nd prime
}}
{{eqn | n = 2
| l = 207 \, 622 \, 297
| r = 4 \times 51 \, 905 \, 574 + 1
| c = and is the $11 \, 477 \, 483$rd prime
}}
{{eqn | n = 3
| l = 20... | The [[Definition:Integer Sequence|sequence]] of $16$ consecutive [[Definition:Prime Number|prime numbers]] beginning from $207 \, 622 \, 273$ are all of the form $4 n + 1$. | {{begin-eqn}}
{{eqn | n = 1
| l = 207 \, 622 \, 273
| r = 4 \times 51 \, 905 \, 568 + 1
| c = and is the $11 \, 477 \, 482$nd [[Definition:Prime Number|prime]]
}}
{{eqn | n = 2
| l = 207 \, 622 \, 297
| r = 4 \times 51 \, 905 \, 574 + 1
| c = and is the $11 \, 477 \, 483$rd [[Definit... | Consecutive Primes of form 4n+1 | https://proofwiki.org/wiki/Consecutive_Primes_of_form_4n+1 | https://proofwiki.org/wiki/Consecutive_Primes_of_form_4n+1 | [
"Prime Numbers"
] | [
"Definition:Integer 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",
"Definition:Prime Number",... |
proofwiki-13677 | Integer which is Sum of 3 Fourth Powers in 2 Ways and Products of Those Roots | The positive integer $256 \, 103 \, 393$ can be expressed as the sum of $3$ fourth powers in $2$ different ways:
{{begin-eqn}}
{{eqn | l = 256 \, 103 \, 393
| r = 22^4 + 93^4 + 116^4
| c =
}}
{{eqn | r = 29^4 + 66^4 + 124^4
| c =
}}
{{end-eqn}}
Also note that:
{{begin-eqn}}
{{eqn | l = 237 \, 336
... | We have that:
{{begin-eqn}}
{{eqn | l = 256 \, 103 \, 393
| r = 234 \, 256 + 74 \, 805 \, 201 + 181 \, 063 \, 936
| c =
}}
{{eqn | r = 22^4 + 93^4 + 116^4
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 256 \, 103 \, 393
| r = 707 \, 281 + 18 \, 974 \, 736 + 236 \, 421 \, 376
| c =
}}
{{... | The [[Definition:Positive Integer|positive integer]] $256 \, 103 \, 393$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Fourth Power|fourth powers]] in $2$ different ways:
{{begin-eqn}}
{{eqn | l = 256 \, 103 \, 393
| r = 22^4 + 93^4 + 116^4
| c =
}}
{{eqn | r = 29^4 + 66^... | We have that:
{{begin-eqn}}
{{eqn | l = 256 \, 103 \, 393
| r = 234 \, 256 + 74 \, 805 \, 201 + 181 \, 063 \, 936
| c =
}}
{{eqn | r = 22^4 + 93^4 + 116^4
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 256 \, 103 \, 393
| r = 707 \, 281 + 18 \, 974 \, 736 + 236 \, 421 \, 376
| c =
}}... | Integer which is Sum of 3 Fourth Powers in 2 Ways and Products of Those Roots | https://proofwiki.org/wiki/Integer_which_is_Sum_of_3_Fourth_Powers_in_2_Ways_and_Products_of_Those_Roots | https://proofwiki.org/wiki/Integer_which_is_Sum_of_3_Fourth_Powers_in_2_Ways_and_Products_of_Those_Roots | [
"Fourth Powers",
"256,103,393"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [] |
proofwiki-13678 | Groups of Order 8 | Let $G$ be a group of order $8$.
Then $G$ is isomorphic to one of the following:
:$\Z_8$
:$\Z_4 \oplus \Z_2$
:$\Z_2 \oplus \Z_2 \oplus \Z_2$
:$D_4$
:$\Dic 2$
where:
:$\Z_n$ is the cyclic group of order $n$
:$D_4$ is the dihedral group of order $8$
:$\Dic 2$ is the dicyclic group of order $8$, also known as the quaterni... | The abelian cases are handled by {{Corollary|Abelian Group Factored by Prime}}.
{{qed|lemma}}
Let $G$ be non-abelian.
By Lagrange's Theorem the order of non-identity elements in $G$ is either $2$, $4$ or $8$.
{{AimForCont}} that there exists an order $8$ element.
Then $G$ is generated by this element.
So $G$ is by defi... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $8$.
Then $G$ is [[Definition:Group Isomorphism|isomorphic]] to one of the following:
:$\Z_8$
:$\Z_4 \oplus \Z_2$
:$\Z_2 \oplus \Z_2 \oplus \Z_2$
:$D_4$
:$\Dic 2$
where:
:$\Z_n$ is the [[Definition:Cyclic Group|cyclic group]] of order $n... | The [[Definition:Abelian Group|abelian]] cases are handled by {{Corollary|Abelian Group Factored by Prime}}.
{{qed|lemma}}
Let $G$ be non-[[Definition:Abelian Group|abelian]].
By [[Lagrange's Theorem (Group Theory)|Lagrange's Theorem]] the [[Definition:Order of Group Element|order]] of non-[[Definition:Identity Elem... | Groups of Order 8 | https://proofwiki.org/wiki/Groups_of_Order_8 | https://proofwiki.org/wiki/Groups_of_Order_8 | [
"Order of Groups",
"Groups of Order 8"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Cyclic Group",
"Definition:Dihedral Group",
"Definition:Dicyclic Group",
"Definition:Dicyclic Group/Quaternion Group"
] | [
"Definition:Abelian Group",
"Definition:Abelian Group",
"Lagrange's Theorem (Group Theory)",
"Definition:Order of Group Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Order of Group Element",
"Definition:Generated Subgroup",
"Definition:Element",
"Definition:Cycli... |
proofwiki-13679 | Triangular Numbers which are Product of 3 Consecutive Integers | The $6$ triangular numbers which can be expressed as the product of $3$ consecutive integers are:
:$6, 120, 210, 990, 185 \, 836, 258 \, 474 \, 216$
{{OEIS|A001219}} | {{begin-eqn}}
{{eqn | l = T_3
| r = \frac {3 \left({3 + 1}\right)} 2
| c = Closed Form for Triangular Numbers
}}
{{eqn | r = 6
| c =
}}
{{eqn | r = 1 \times 2 \times 3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = T_{15}
| r = \frac {15 \left({15 + 1}\right)} 2
| c = Closed Form f... | The $6$ [[Definition:Triangular Number|triangular numbers]] which can be expressed as the [[Definition:Integer Multiplication|product]] of $3$ consecutive [[Definition:Integer|integers]] are:
:$6, 120, 210, 990, 185 \, 836, 258 \, 474 \, 216$
{{OEIS|A001219}} | {{begin-eqn}}
{{eqn | l = T_3
| r = \frac {3 \left({3 + 1}\right)} 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | r = 6
| c =
}}
{{eqn | r = 1 \times 2 \times 3
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = T_{15}
| r = \frac {15 \left({15 + 1}\right)} 2
| c = [[Close... | Triangular Numbers which are Product of 3 Consecutive Integers | https://proofwiki.org/wiki/Triangular_Numbers_which_are_Product_of_3_Consecutive_Integers | https://proofwiki.org/wiki/Triangular_Numbers_which_are_Product_of_3_Consecutive_Integers | [
"Triangular Numbers"
] | [
"Definition:Triangular Number",
"Definition:Multiplication/Integers",
"Definition:Integer"
] | [
"Closed Form for Triangular Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Triangular Numbers",
"Closed Form for Triangular Numbers"
] |
proofwiki-13680 | First Harmonic Number to exceed 20 | The first harmonic number that is greater than $20$ is $H_{272 \, 400 \, 600}$.
That is, the number of terms of the harmonic series required for its partial sum to exceed $20$ is $272 \, 400 \, 600$. | We have:
:$H_{272 \, 400 \, 599} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 599} \frac 1 k \approx 19 \cdotp 99999 \, 99979$
and:
:$H_{272 \, 400 \, 600} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 600} \frac 1 k \approx 20 \cdotp 00000 \, 00016$ | The first [[Definition:Harmonic Number|harmonic number]] that is greater than $20$ is $H_{272 \, 400 \, 600}$.
That is, the number of [[Definition:Term of Sequence|terms]] of the [[Definition:Harmonic Series|harmonic series]] required for its [[Definition:Partial Sum|partial sum]] to exceed $20$ is $272 \, 400 \, 600$... | We have:
:$H_{272 \, 400 \, 599} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 599} \frac 1 k \approx 19 \cdotp 99999 \, 99979$
and:
:$H_{272 \, 400 \, 600} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 600} \frac 1 k \approx 20 \cdotp 00000 \, 00016$ | First Harmonic Number to exceed 20 | https://proofwiki.org/wiki/First_Harmonic_Number_to_exceed_20 | https://proofwiki.org/wiki/First_Harmonic_Number_to_exceed_20 | [
"Harmonic Numbers",
"272,400,600"
] | [
"Definition:Harmonic Numbers",
"Definition:Term of Sequence",
"Definition:Harmonic Series",
"Definition:Series/Sequence of Partial Sums"
] | [] |
proofwiki-13681 | First Harmonic Number to exceed 10 | The first harmonic number that is greater than $10$ is $H_{12 \, 367}$.
That is, the number of terms of the harmonic series required for its partial sum to exceed $10$ is $12 \, 367$. | We have:
:$H_{12 \, 366} = \ds \sum_{k \mathop = 1}^{12 \, 366} \frac 1 k \approx 9 \cdotp 99996 \, 214$
and:
:$H_{12 \, 367} = \ds \sum_{k \mathop = 1}^{12 \, 367} \frac 1 k \approx 10 \cdotp 00004 \, 30083$ | The first [[Definition:Harmonic Number|harmonic number]] that is greater than $10$ is $H_{12 \, 367}$.
That is, the number of [[Definition:Term of Sequence|terms]] of the [[Definition:Harmonic Series|harmonic series]] required for its [[Definition:Partial Sum|partial sum]] to exceed $10$ is $12 \, 367$. | We have:
:$H_{12 \, 366} = \ds \sum_{k \mathop = 1}^{12 \, 366} \frac 1 k \approx 9 \cdotp 99996 \, 214$
and:
:$H_{12 \, 367} = \ds \sum_{k \mathop = 1}^{12 \, 367} \frac 1 k \approx 10 \cdotp 00004 \, 30083$ | First Harmonic Number to exceed 10 | https://proofwiki.org/wiki/First_Harmonic_Number_to_exceed_10 | https://proofwiki.org/wiki/First_Harmonic_Number_to_exceed_10 | [
"Harmonic Numbers",
"12,367"
] | [
"Definition:Harmonic Numbers",
"Definition:Term of Sequence",
"Definition:Harmonic Series",
"Definition:Series/Sequence of Partial Sums"
] | [] |
proofwiki-13682 | Canonical Homomorphism to Polynomial Ring is Ring Monomorphism | Let $R$ be a commutative ring with unity.
Let $\struct {R \sqbrk X, \iota, X}$ be a polynomial ring over $R$ in one indeterminate $X$.
Then the canonical homomorphism $\iota : R \to R \sqbrk X$ is a ring monomorphism. | Let $\operatorname{id} : R \to R$ be the identity mapping.
Let $1$ be the unity of $R$.
By Identity Mapping is Ring Automorphism, $\operatorname{id}$ is a ring homomorphism.
By Universal Property of Polynomial Ring, there exists a ring homomorphism $h : R \sqbrk X \to R$ with $h \circ \iota = \operatorname{id}$.
By Ide... | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\struct {R \sqbrk X, \iota, X}$ be a [[Definition:Polynomial Ring in one Indeterminate|polynomial ring]] over $R$ in one [[Definition:Indeterminate of Polynomial Ring|indeterminate]] $X$.
Then the [[Definition:Embedding into Po... | Let $\operatorname{id} : R \to R$ be the [[Definition:Identity Mapping|identity mapping]].
Let $1$ be the [[Definition:Unity of Ring|unity]] of $R$.
By [[Identity Mapping is Ring Automorphism]], $\operatorname{id}$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
By [[Universal Property of Polynomial Ring]],... | Canonical Homomorphism to Polynomial Ring is Ring Monomorphism | https://proofwiki.org/wiki/Canonical_Homomorphism_to_Polynomial_Ring_is_Ring_Monomorphism | https://proofwiki.org/wiki/Canonical_Homomorphism_to_Polynomial_Ring_is_Ring_Monomorphism | [
"Polynomial Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Polynomial Ring/Embedding",
"Definition:Ring Monomorphism"
] | [
"Definition:Identity Mapping",
"Definition:Unity (Abstract Algebra)/Ring",
"Identity Mapping is Automorphism/Rings",
"Definition:Ring Homomorphism",
"Universal Property of Polynomial Ring",
"Definition:Ring Homomorphism",
"Identity Mapping is Injection",
"Definition:Injection",
"Injection if Composi... |
proofwiki-13683 | Universal Property of Field of Rational Fractions | Let $R$ be an integral domain.
Let $\struct {\map R x, \iota, x}$ be the field of rational fractions over $R$.
Let $\struct {K, f, a}$ be an ordered triple, where:
:$K$ is a field
:$f : R \to K$ is a unital ring homomorphism
:$a$ is a transcendental element of $K$.
Then there exists a unique unital ring homomorphism $\... | Use Universal Property of Polynomial Ring and Universal Poperty of Field of Fractions.
{{ProofWanted}}
Category:Polynomial Theory
Category:Universal Properties
hj182iw7n4d1b25r40gnv4qd4f2848d | Let $R$ be an [[Definition:Integral Domain|integral domain]].
Let $\struct {\map R x, \iota, x}$ be the [[Definition:Field of Rational Fractions|field of rational fractions]] over $R$.
Let $\struct {K, f, a}$ be an [[Definition:Ordered Triple|ordered triple]], where:
:$K$ is a [[Definition:Field (Abstract Algebra)|fi... | Use [[Universal Property of Polynomial Ring]] and [[Universal Poperty of Field of Fractions]].
{{ProofWanted}}
[[Category:Polynomial Theory]]
[[Category:Universal Properties]]
hj182iw7n4d1b25r40gnv4qd4f2848d | Universal Property of Field of Rational Fractions | https://proofwiki.org/wiki/Universal_Property_of_Field_of_Rational_Fractions | https://proofwiki.org/wiki/Universal_Property_of_Field_of_Rational_Fractions | [
"Polynomial Theory",
"Universal Properties"
] | [
"Definition:Integral Domain",
"Definition:Field of Rational Fractions",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Field (Abstract Algebra)",
"Definition:Unital Ring Homomorphism",
"Definition:Transcendental Element of Algebra over Ring",
"Definition:Unique",
"Definition:Uni... | [
"Universal Property of Polynomial Ring",
"Universal Poperty of Field of Fractions",
"Category:Polynomial Theory",
"Category:Universal Properties"
] |
proofwiki-13684 | Equivalence of Definitions of Unital Subalgebra | Let $R$ be a commutative ring.
Let $\struct {A_R, *}$ be an unital algebra over $R$ whose unit is $1_A$.
Let $\struct {B_R, *}$ be a subalgebra of $A_R$.
{{TFAE|def = Unital Subalgebra}} | {{ProofWanted}}
Category:Unital Subalgebras
3hdv92dy2uskggb8ofcf5dd6n3aobr7 | Let $R$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {A_R, *}$ be an [[Definition:Unital Algebra|unital algebra]] over $R$ whose [[Definition:Unit of Algebra|unit]] is $1_A$.
Let $\struct {B_R, *}$ be a [[Definition:Subalgebra|subalgebra]] of $A_R$.
{{TFAE|def = Unital Subalgebra}} | {{ProofWanted}}
[[Category:Unital Subalgebras]]
3hdv92dy2uskggb8ofcf5dd6n3aobr7 | Equivalence of Definitions of Unital Subalgebra | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Unital_Subalgebra | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Unital_Subalgebra | [
"Unital Subalgebras"
] | [
"Definition:Commutative Ring",
"Definition:Unital Algebra",
"Definition:Unit of Algebra",
"Definition:Subalgebra"
] | [
"Category:Unital Subalgebras"
] |
proofwiki-13685 | Polydivisible Number/Examples/381,654,729 | The integer $381 \, 654 \, 729$ is the only polydivisible number which is penholodigital. | First it is demonstrated that indeed $381 \, 654 \, 729$ has this property:
{{begin-eqn}}
{{eqn | l = 3
| r = 1 \times 3
}}
{{eqn | l = 38
| r = 2 \times 19
}}
{{eqn | l = 381
| r = 3 \times 127
}}
{{eqn | l = 3816
| r = 4 \times 954
}}
{{eqn | l = 38 \, 165
| r = 5 \times 7633
}}
{{eqn | ... | The [[Definition:Integer|integer]] $381 \, 654 \, 729$ is the only [[Definition:Polydivisible Number|polydivisible number]] which is [[Definition:Penholodigital Integer|penholodigital]]. | First it is demonstrated that indeed $381 \, 654 \, 729$ has this property:
{{begin-eqn}}
{{eqn | l = 3
| r = 1 \times 3
}}
{{eqn | l = 38
| r = 2 \times 19
}}
{{eqn | l = 381
| r = 3 \times 127
}}
{{eqn | l = 3816
| r = 4 \times 954
}}
{{eqn | l = 38 \, 165
| r = 5 \times 7633
}}
{{eqn |... | Polydivisible Number/Examples/381,654,729 | https://proofwiki.org/wiki/Polydivisible_Number/Examples/381,654,729 | https://proofwiki.org/wiki/Polydivisible_Number/Examples/381,654,729 | [
"Polydivisible Numbers",
"Penholodigital Integers",
"381,654,729"
] | [
"Definition:Integer",
"Definition:Polydivisible Number",
"Definition:Pandigital Set/Penholodigital/Integer"
] | [
"Definition:Polydivisible Number",
"Definition:Pandigital Set/Penholodigital/Integer",
"Divisibility by 5",
"Divisibility by 2",
"Definition:Even Integer",
"Definition:Odd Integer",
"Divisibility by 9/Corollary",
"Definition:Divisor (Algebra)/Integer",
"Divisibility by 8",
"Definition:Divisor (Alg... |
proofwiki-13686 | Pandigital Product of Pandigital Pairs in 3 Ways | The pandigital integer $0 \, 429 \, 315 \, 678$ can be expressed as the product of a pandigital doubleton in $3$ different ways:
{{begin-eqn}}
{{eqn | l = 0 \, 429 \, 315 \, 678
| r = 04 \, 926 \times 87 \, 153
}}
{{eqn | r = 07 \, 923 \times 54 \, 186}}
{{eqn | r = 15 \, 846 \times 27 \, 093}}
{{end-eqn}} | We have that:
:$0 \, 429 \, 315 \, 678 = 2 \times 3^2 \times 11 \times 19 \times 139 \times 821$
Then:
{{begin-eqn}}
{{eqn | l = 04 \, 926 \times 87 \, 153
| r = \paren {2 \times 3 \times 821} \times \paren {3 \times 11 \times 19 \times 139}
}}
{{eqn | l = 07 \, 923 \times 54 \, 186
| r = \paren {3 \times 1... | The [[Definition:Pandigital Integer|pandigital integer]] $0 \, 429 \, 315 \, 678$ can be expressed as the [[Definition:Integer Multiplication|product]] of a [[Definition:Pandigital Set|pandigital]] [[Definition:Doubleton|doubleton]] in $3$ different ways:
{{begin-eqn}}
{{eqn | l = 0 \, 429 \, 315 \, 678
| r = 0... | We have that:
:$0 \, 429 \, 315 \, 678 = 2 \times 3^2 \times 11 \times 19 \times 139 \times 821$
Then:
{{begin-eqn}}
{{eqn | l = 04 \, 926 \times 87 \, 153
| r = \paren {2 \times 3 \times 821} \times \paren {3 \times 11 \times 19 \times 139}
}}
{{eqn | l = 07 \, 923 \times 54 \, 186
| r = \paren {3 \times... | Pandigital Product of Pandigital Pairs in 3 Ways | https://proofwiki.org/wiki/Pandigital_Product_of_Pandigital_Pairs_in_3_Ways | https://proofwiki.org/wiki/Pandigital_Product_of_Pandigital_Pairs_in_3_Ways | [
"Pandigital Sets"
] | [
"Definition:Pandigital Set/Integer",
"Definition:Multiplication/Integers",
"Definition:Pandigital Set",
"Definition:Doubleton"
] | [] |
proofwiki-13687 | Smallest Positive Integer which is Sum of 2 Fourth Powers in 2 Ways | The smallest positive integer which can be expressed as the sum of $2$ fourth powers in $2$ different ways is:
{{begin-eqn}}
{{eqn | l = 635 \, 318 \, 657
| r = 59^4 + 158^4
| c =
}}
{{eqn | r = 133^4 + 134^4
| c =
}}
{{end-eqn}} | The fact that these are the smallest can be demonstrated by calculation.
{{qed}} | The smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Fourth Power|fourth powers]] in $2$ different ways is:
{{begin-eqn}}
{{eqn | l = 635 \, 318 \, 657
| r = 59^4 + 158^4
| c =
}}
{{eqn | r = 133^4 + 134^4
... | The fact that these are the smallest can be demonstrated by calculation.
{{qed}} | Smallest Positive Integer which is Sum of 2 Fourth Powers in 2 Ways | https://proofwiki.org/wiki/Smallest_Positive_Integer_which_is_Sum_of_2_Fourth_Powers_in_2_Ways | https://proofwiki.org/wiki/Smallest_Positive_Integer_which_is_Sum_of_2_Fourth_Powers_in_2_Ways | [
"Fourth Powers",
"635,318,657"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Fourth Power"
] | [] |
proofwiki-13688 | Largest Penholodigital Square | The largest penholodigital square is $923 \, 187 \, 456$:
:$923 \, 187 \, 456 = 30 \, 384^2$ | {{ProofWanted|Needs to be demonstrated that there are none higher. Could be done by checking all the squares from $30 \, 385^2$ up to $31 \, 426$ but that's too boring for now.}} | The largest [[Definition:Penholodigital Integer|penholodigital]] [[Definition:Square Number|square]] is $923 \, 187 \, 456$:
:$923 \, 187 \, 456 = 30 \, 384^2$ | {{ProofWanted|Needs to be demonstrated that there are none higher. Could be done by checking all the squares from $30 \, 385^2$ up to $31 \, 426$ but that's too boring for now.}} | Largest Penholodigital Square | https://proofwiki.org/wiki/Largest_Penholodigital_Square | https://proofwiki.org/wiki/Largest_Penholodigital_Square | [
"Square Numbers",
"Penholodigital Integers",
"923,187,456"
] | [
"Definition:Pandigital Set/Penholodigital/Integer",
"Definition:Square Number"
] | [] |
proofwiki-13689 | Largest 9-Digit Prime Number | The largest prime number with $9$ digits is $999 \, 999 \, 937$. | Consider the numbers $\sqbrk {999 \, 999 \, 9ab}$.
Since $999 \, 999 \, 000$ is divisible by $2, 3, 5, 7, 11, 13$,
if $\sqbrk {9ab}$ is divisible by these primes, so is $\sqbrk {999 \, 999 \, 9ab}$.
After this elimination the only $\sqbrk {ab} > 37$ that remains are:
:$41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97$
We... | The largest [[Definition:Prime Number|prime number]] with $9$ [[Definition:Digit|digits]] is $999 \, 999 \, 937$. | Consider the numbers $\sqbrk {999 \, 999 \, 9ab}$.
Since $999 \, 999 \, 000$ is [[Definition:Divisor|divisible]] by $2, 3, 5, 7, 11, 13$,
if $\sqbrk {9ab}$ is [[Definition:Divisor|divisible]] by these [[Definition:Prime Number|primes]], so is $\sqbrk {999 \, 999 \, 9ab}$.
After this elimination the only $\sqbrk {ab}... | Largest 9-Digit Prime Number | https://proofwiki.org/wiki/Largest_9-Digit_Prime_Number | https://proofwiki.org/wiki/Largest_9-Digit_Prime_Number | [
"Specific Numbers",
"999,999,937"
] | [
"Definition:Prime Number",
"Definition:Digit"
] | [
"Definition:Divisor",
"Definition:Divisor",
"Definition:Prime Number",
"Definition:Prime Number"
] |
proofwiki-13690 | Smallest Pandigital Square | The smallest pandigital square is $1 \, 026 \, 753 \, 849$:
:$1 \, 026 \, 753 \, 849 = 32 \, 043^2$ | We check all the squares of numbers from $\ceiling {\sqrt {1 \, 023 \, 456 \, 789} } = 31 \, 992$ up to $32 \, 042$, with the following constraints:
Since all these squares has $10$ as its two leftmost digits, the number cannot end with $0$, $1$ or $9$.
A pandigital number is divisible by $9$, so our number must be div... | The smallest [[Definition:Pandigital Integer|pandigital]] [[Definition:Square Number|square]] is $1 \, 026 \, 753 \, 849$:
:$1 \, 026 \, 753 \, 849 = 32 \, 043^2$ | We check all the [[Definition:Square Number|squares]] of numbers from $\ceiling {\sqrt {1 \, 023 \, 456 \, 789} } = 31 \, 992$ up to $32 \, 042$, with the following constraints:
Since all these [[Definition:Square Number|squares]] has $10$ as its two leftmost [[Definition:Digit|digits]], the number cannot end with $0... | Smallest Pandigital Square | https://proofwiki.org/wiki/Smallest_Pandigital_Square | https://proofwiki.org/wiki/Smallest_Pandigital_Square | [
"Square Numbers",
"Pandigital Integers",
"1,026,753,849"
] | [
"Definition:Pandigital Set/Integer",
"Definition:Square Number"
] | [
"Definition:Square Number",
"Definition:Square Number",
"Definition:Digit",
"Definition:Pandigital Set/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Pandigital Set/Integer"
] |
proofwiki-13691 | Sound Proof System is Consistent | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\mathscr P$ be a proof system for $\LL$.
Suppose that $\mathscr P$ is sound for $\mathscr M$.
Then $\mathscr P$ is consistent. | By assumption, some logical formula $\phi$ is not an $\mathscr M$-tautology.
Since $\mathscr P$ is sound for $\mathscr M$, $\phi$ is also not a $\mathscr P$-theorem.
But then by definition $\mathscr P$ is consistent.
{{qed}} | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\mathscr P$ be a [[Definition:Proof System|proof system]] for $\LL$.
Suppose that $\mathscr P$ is [[Definition:Sound Proof System|sound]] for $\mathscr M$.
Then $\... | By assumption, some [[Definition:Logical Formula|logical formula]] $\phi$ is not an $\mathscr M$-[[Definition:Tautology (Formal Semantics)|tautology]].
Since $\mathscr P$ is [[Definition:Sound Proof System|sound]] for $\mathscr M$, $\phi$ is also not a $\mathscr P$-[[Definition:Theorem (Formal Systems)|theorem]].
But... | Sound Proof System is Consistent | https://proofwiki.org/wiki/Sound_Proof_System_is_Consistent | https://proofwiki.org/wiki/Sound_Proof_System_is_Consistent | [
"Proof Systems"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Proof System",
"Definition:Sound Proof System",
"Definition:Consistent (Logic)/Proof System"
] | [
"Definition:Logical Formula",
"Definition:Tautology/Formal Semantics",
"Definition:Sound Proof System",
"Definition:Theorem/Formal System",
"Definition:Consistent (Logic)/Proof System"
] |
proofwiki-13692 | Equivalence of Definitions of Consistent Proof System | {{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}}
Let $\LL_0$ be the language of propositional logic.
Let $\mathscr P$ be a proof system for $\LL_0$. | === Definition 1 implies Definition 2 ===
Suppose that $\neg \vdash_{\mathscr P} \phi$.
Suppose additionally that there is some logical formula $\psi$ such that:
:$\vdash_{\mathscr P} \psi, \neg \psi$
By the Rule of Explosion:
:$\psi, \neg \psi \vdash_{\mathscr P} \phi$
By Provable Consequence of Theorems is Theorem, w... | {{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}}
Let $\LL_0$ be the [[Definition:Language of Propositional Logic|language of propositional logic]].
Let $\mathscr P$ be a [[Definition:Proof System|proof system]] for $\LL_0$. | === Definition 1 implies Definition 2 ===
Suppose that $\neg \vdash_{\mathscr P} \phi$.
Suppose additionally that there is some [[Definition:Logical Formula|logical formula]] $\psi$ such that:
:$\vdash_{\mathscr P} \psi, \neg \psi$
By the [[Rule of Explosion/Variant 3|Rule of Explosion]]:
:$\psi, \neg \psi \vdash_... | Equivalence of Definitions of Consistent Proof System | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Consistent_Proof_System | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Consistent_Proof_System | [
"Proof Systems",
"Propositional Logic"
] | [
"Definition:Language of Propositional Logic",
"Definition:Proof System"
] | [
"Definition:Logical Formula",
"Rule of Explosion/Variant 3",
"Provable Consequence of Theorems is Theorem"
] |
proofwiki-13693 | Rule of Explosion/Variant 3 | :$p, \neg p \vdash q$ | {{BeginTableau|p, \neg p \vdash q|Instance 2 of the Hilbert-style systems}}
{{Assumption|1|p}}
{{Assumption|2|\neg p}}
{{TableauLine
|n = 3
|f = q \implies (p \lor q)
|rlnk = Definition:Hilbert Proof System/Instance 2
|rtxt = Axiom $A2$
}}
{{TableauLine
|n = 4
|f = \neg p \implies (q \lor \neg p)
|rlnk = Definit... | :$p, \neg p \vdash q$ | {{BeginTableau|p, \neg p \vdash q|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
{{Assumption|1|p}}
{{Assumption|2|\neg p}}
{{TableauLine
|n = 3
|f = q \implies (p \lor q)
|rlnk = Definition:Hilbert Proof System/Instance 2
|rtxt = Axiom $A2$
}}
{{TableauLine
|n = 4
|f = \... | Rule of Explosion/Variant 3 | https://proofwiki.org/wiki/Rule_of_Explosion/Variant_3 | https://proofwiki.org/wiki/Rule_of_Explosion/Variant_3 | [
"Rule of Explosion"
] | [] | [
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-13694 | Smallest Integer which is Sum of 3 Fifth Powers in 2 Ways | The smallest positive integer which can be expressed as the sum of $3$ fifth powers in $2$ different ways:
The positive integer $1 \, 375 \, 298 \, 099$ can be expressed as the sum of $3$ fifth powers in $2$ different ways:
{{begin-eqn}}
{{eqn | l = 1 \, 375 \, 298 \, 099
| r = 24^5 + 28^5 + 67^5
| c =
}}
... | {{begin-eqn}}
{{eqn | l = 1 \, 375 \, 298 \, 099
| r = 7 \, 962 \, 624 + 17 \, 210 \, 368 + 1 \, 350 \, 125 \, 107
| c =
}}
{{eqn | r = 24^5 + 28^5 + 67^5
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1 \, 375 \, 298 \, 099
| r = 243 + 459 \, 165 \, 024 + 916 \, 132 \, 832
| c =
}}
{{e... | The smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Fifth Power|fifth powers]] in $2$ different ways:
The [[Definition:Positive Integer|positive integer]] $1 \, 375 \, 298 \, 099$ can be expressed as the [[Definition:Intege... | {{begin-eqn}}
{{eqn | l = 1 \, 375 \, 298 \, 099
| r = 7 \, 962 \, 624 + 17 \, 210 \, 368 + 1 \, 350 \, 125 \, 107
| c =
}}
{{eqn | r = 24^5 + 28^5 + 67^5
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 1 \, 375 \, 298 \, 099
| r = 243 + 459 \, 165 \, 024 + 916 \, 132 \, 832
| c =
}}
{... | Smallest Integer which is Sum of 3 Fifth Powers in 2 Ways | https://proofwiki.org/wiki/Smallest_Integer_which_is_Sum_of_3_Fifth_Powers_in_2_Ways | https://proofwiki.org/wiki/Smallest_Integer_which_is_Sum_of_3_Fifth_Powers_in_2_Ways | [
"Fifth Powers",
"1,375,298,099"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Fifth Power",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Fifth Power"
] | [] |
proofwiki-13695 | Automorphic Numbers with 10 Digits | The only $10$-digit automorphic numbers are:
:$1 \, 787 \, 109 \, 376$
:$8 \, 212 \, 890 \, 625$ | We have:
{{begin-eqn}}
{{eqn | l = 1 \, 787 \, 109 \, 376^2
| r = \enspace 3 \, 193 \, 759 \, 92 \mathbf {1 \, 787 \, 109 \, 376}
}}
{{eqn | l = 8 \, 212 \, 890 \, 625^2
| r = 67 \, 451 \, 572 \, 41 \mathbf {8 \, 212 \, 890 \, 625}
}}
{{end-eqn}}
thus demonstrating they are automorphic.
By Automorphic Numbe... | The only $10$-[[Definition:Digit|digit]] [[Definition:Automorphic Number|automorphic numbers]] are:
:$1 \, 787 \, 109 \, 376$
:$8 \, 212 \, 890 \, 625$ | We have:
{{begin-eqn}}
{{eqn | l = 1 \, 787 \, 109 \, 376^2
| r = \enspace 3 \, 193 \, 759 \, 92 \mathbf {1 \, 787 \, 109 \, 376}
}}
{{eqn | l = 8 \, 212 \, 890 \, 625^2
| r = 67 \, 451 \, 572 \, 41 \mathbf {8 \, 212 \, 890 \, 625}
}}
{{end-eqn}}
thus demonstrating they are [[Definition:Automorphic Number|... | Automorphic Numbers with 10 Digits | https://proofwiki.org/wiki/Automorphic_Numbers_with_10_Digits | https://proofwiki.org/wiki/Automorphic_Numbers_with_10_Digits | [
"Automorphic Numbers"
] | [
"Definition:Digit",
"Definition:Automorphic Number"
] | [
"Definition:Automorphic Number",
"Automorphic Numbers in Base 10"
] |
proofwiki-13696 | Left-Truncated Automorphic Number is Automorphic | Let $n$ be an automorphic number, expressed in some conventional number base.
Let any number of digits be removed from the left-hand end of $n$.
Then what remains is also an automorphic number. | Let $n$ be an automorphic number of $d$ digits, expressed in base $b$.
By {{Defof|Automorphic Number}}, we have:
:$n^2 \equiv n \pmod {b^d}$
Let some digits be removed from the left-hand end of $n$, so that only $d'$ digits remain.
This only makes sense when $d' < d$.
Define this new number as $n'$.
Then we have:
:$n \... | Let $n$ be an [[Definition:Automorphic Number|automorphic number]], expressed in some conventional [[Definition:Number Base|number base]].
Let any number of [[Definition:Digit|digits]] be removed from the left-hand end of $n$.
Then what remains is also an [[Definition:Automorphic Number|automorphic number]]. | Let $n$ be an [[Definition:Automorphic Number|automorphic number]] of $d$ [[Definition:Digit|digits]], expressed in [[Definition:Number Base|base]] $b$.
By {{Defof|Automorphic Number}}, we have:
:$n^2 \equiv n \pmod {b^d}$
Let some [[Definition:Digit|digits]] be removed from the left-hand end of $n$, so that only $... | Left-Truncated Automorphic Number is Automorphic | https://proofwiki.org/wiki/Left-Truncated_Automorphic_Number_is_Automorphic | https://proofwiki.org/wiki/Left-Truncated_Automorphic_Number_is_Automorphic | [
"Automorphic Numbers",
"Left-Truncated Automorphic Number is Automorphic"
] | [
"Definition:Automorphic Number",
"Definition:Number Base",
"Definition:Digit",
"Definition:Automorphic Number"
] | [
"Definition:Automorphic Number",
"Definition:Digit",
"Definition:Number Base",
"Definition:Digit",
"Congruence by Divisor of Modulus",
"Congruence of Powers",
"Definition:Automorphic Number",
"Definition:Digit",
"Definition:Number Base"
] |
proofwiki-13697 | Square whose Divisor Sum is Cubic | The number $1 \, 857 \, 437 \, 604$ is a square number whose divisor sum is a cube. | {{begin-eqn}}
{{eqn | l = 1 \, 857 \, 437 \, 604
| r = 43 \, 098^2
| c =
}}
{{eqn | l = \map {\sigma_1} {1 \, 857 \, 437 \, 604}
| r = 5 \, 168 \, 743 \, 489
| c = {{DSFLink|1,857,437,604|1 \, 857 \, 437 \, 604}}
}}
{{eqn | r = 1729^3
| c =
}}
{{end-eqn}}
{{qed}} | The number $1 \, 857 \, 437 \, 604$ is a [[Definition:Square Number|square number]] whose [[Definition:Divisor Sum Function|divisor sum]] is a [[Definition:Cube Number|cube]]. | {{begin-eqn}}
{{eqn | l = 1 \, 857 \, 437 \, 604
| r = 43 \, 098^2
| c =
}}
{{eqn | l = \map {\sigma_1} {1 \, 857 \, 437 \, 604}
| r = 5 \, 168 \, 743 \, 489
| c = {{DSFLink|1,857,437,604|1 \, 857 \, 437 \, 604}}
}}
{{eqn | r = 1729^3
| c =
}}
{{end-eqn}}
{{qed}} | Square whose Divisor Sum is Cubic | https://proofwiki.org/wiki/Square_whose_Divisor_Sum_is_Cubic | https://proofwiki.org/wiki/Square_whose_Divisor_Sum_is_Cubic | [
"Divisor Sum Function",
"Square Numbers",
"Cube Numbers",
"1,857,437,604",
"Integers whose Divisor Sum is Cube"
] | [
"Definition:Square Number",
"Definition:Divisor Sum Function",
"Definition:Cube Number"
] | [] |
proofwiki-13698 | Largest Right-Truncatable Primes allowing 1 | Let $1$ be temporarily considered to be a prime number.
Under that consideration, the largest right-truncatable prime numbers are:
:$1 \, 979 \, 339 \, 333$
:$1 \, 979 \, 339 \, 339$ | We have that:
{{begin-eqn}}
{{eqn | o =
| r = 1 \, 979 \, 339 \, 333
| c = is prime
}}
{{eqn | o =
| r = 1 \, 979 \, 339 \, 339
| c = is prime
}}
{{end-eqn}}
For both, the truncation process is the same:
{{begin-eqn}}
{{eqn | o =
| r = 197 \, 933 \, 933
| c = is the $10 \, 970 \, ... | Let $1$ be temporarily considered to be a [[Definition:Prime Number|prime number]].
Under that consideration, the largest [[Definition:Right-Truncatable Prime|right-truncatable]] [[Definition:Prime Number|prime numbers]] are:
:$1 \, 979 \, 339 \, 333$
:$1 \, 979 \, 339 \, 339$ | We have that:
{{begin-eqn}}
{{eqn | o =
| r = 1 \, 979 \, 339 \, 333
| c = is [[Definition:Prime Number|prime]]
}}
{{eqn | o =
| r = 1 \, 979 \, 339 \, 339
| c = is [[Definition:Prime Number|prime]]
}}
{{end-eqn}}
For both, the truncation process is the same:
{{begin-eqn}}
{{eqn | o =
... | Largest Right-Truncatable Primes allowing 1 | https://proofwiki.org/wiki/Largest_Right-Truncatable_Primes_allowing_1 | https://proofwiki.org/wiki/Largest_Right-Truncatable_Primes_allowing_1 | [
"Right-Truncatable Primes"
] | [
"Definition:Prime Number",
"Definition:Right-Truncatable Prime",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number"
... |
proofwiki-13699 | Completely Multiplicative Function is Multiplicative | Let $f: \Z \to \Z$ be a function on the integers $\Z$.
Let $f$ be completely multiplicative.
{{Questionable|Complete multiplicativity is defined for fields, but $\Z$ is not a field.}}
Then $f$ is multiplicative.
{{Questionable|Multiplicativity is defined for $f : \N \to \N$, undefined for $f : \Z \to \Z$}} | By definition of complete multiplicativity:
:$\forall m, n \in \Z: \map f {m n} = \map f m \map f n$
Hence by True Statement is implied by Every Statement:
:$\forall m, n \in \Z: m \perp n \implies \map f {m n} = \map f m \map f n$
So $f$ is multiplicative.
{{qed}}
Category:Number Theory
Category:Completely Multiplicat... | Let $f: \Z \to \Z$ be a [[Definition:Function|function]] on the [[Definition:Integer|integers]] $\Z$.
Let $f$ be [[Definition:Completely Multiplicative Function|completely multiplicative]].
{{Questionable|Complete multiplicativity is defined for fields, but $\Z$ is not a field.}}
Then $f$ is [[Definition:Multiplicat... | By definition of [[Definition:Completely Multiplicative Function|complete multiplicativity]]:
:$\forall m, n \in \Z: \map f {m n} = \map f m \map f n$
Hence by [[True Statement is implied by Every Statement]]:
:$\forall m, n \in \Z: m \perp n \implies \map f {m n} = \map f m \map f n$
So $f$ is [[Definition:Multipl... | Completely Multiplicative Function is Multiplicative | https://proofwiki.org/wiki/Completely_Multiplicative_Function_is_Multiplicative | https://proofwiki.org/wiki/Completely_Multiplicative_Function_is_Multiplicative | [
"Number Theory",
"Completely Multiplicative Functions",
"Multiplicative Functions"
] | [
"Definition:Function",
"Definition:Integer",
"Definition:Completely Multiplicative Function",
"Definition:Multiplicative Arithmetic Function"
] | [
"Definition:Completely Multiplicative Function",
"True Statement is implied by Every Statement",
"Definition:Multiplicative Arithmetic Function",
"Category:Number Theory",
"Category:Completely Multiplicative Functions",
"Category:Multiplicative Functions"
] |
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