hcju/proofwikips huggingface dataset:
Smallest Integer which is Product of 4 Triples all with Same Sum
The smallest integer which can be expressed as the product of $4$ different triplets of integers each of which has the same sum is: {{begin-eqn}} {{eqn | l = 25 \, 200       | r = 6 \times 56 \times 75 }} {{eqn | r = 7 \times 40 \times 90 }} {{eqn | r = 9 \times 28 \times 100 }} {{eqn | r = 12 \times 20 \times 105 }} {{end-eqn}}
4 Integers whose Euler Phi Value is 10,368
:$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 10 \, 368 = 2^7 \times 3^4$ where $\phi$ denotes the Euler $\phi$ function.
Change of Lead in Prime Number Race 4n+1 vs. 4n-1
Consider the prime number race between $4 n + 1$ and $4 n - 1$. While the prime numbers of the form $4 n - 1$ appear usually to be in the majority, the lead changes from one to the other an infinite number of times.
Fourth Power expressible as Sum of 6 Fourth Powers
$28 \, 561$ can be expressed as the sum of $6$ fourth powers: :$28 \, 561 = 13^4 = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4$
Smallest Differences between Fractional Parts of Square and Cube Roots
Apart from $6$th powers, the value of $n$ less than $50 \, 000$ for which the difference between the fractional parts of $\sqrt n$ and $\sqrt [3] n$ is smallest is $30 \, 739$. The next integer to produce a smaller difference above that is $62 \, 324$.
Smallest Triplet of Consecutive Integers each Divisible by Fourth Power
This triplet of consecutive integers has the property that each of them is divisible by a fourth power: :$33 \, 614, 33 \, 615, 33 \, 616$ This is the smallest such triplet.
Abundancy of Integers in form 945 + 630n
A large number of integers of the form $945 + 630 n$, for $n \in \Z_{\ge 0}$, are abundant. The first counterexample is for $n = 52$.
Sigma Function of Integer/Corollary
:$\displaystyle \map \sigma n = \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop > 1} } \frac {p_i^{k_i + 1} - 1} {p_i - 1} \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop = 1} } \paren {p_i + 1}$
Affine Group of One Dimension as Semidirect Product
Let $\map {\operatorname{Af}_1} \R$ be the $1$-dimensional affine group on $\R$. Let $\R^+$ be the additive group of real numbers. Let $\R^\times$ be the multiplicative group of real numbers. Let $\phi: \R^\times \to \Aut {\R^+}$ be defined as: :$\forall b \in \R^\times: \map \phi b = \paren {a \mapsto a b}$ Let $\R^+ \rtimes_\phi \R^\times$ be the corresponding semidirect product. Then: :$\map {\operatorname {Af}_1} \R \cong \R^+ \rtimes_\phi \R^\times$ where $\cong$ denotes (group) isomorphism.
Sequence of Consecutive Integers with Same Number of Divisors
The following sequence of consecutive integers all have the same number of divisors, that is, $8$: :$40 \, 311, 40 \, 312, 40 \, 313, 40 \, 314, 40 \, 315$ This is the longest such sequence known.
40,585/Historical Note
The fact that $40 \, 585$ is a factorion base $10$  was discovered as late as $1964$ by {{AuthorRef|Leigh Janes}}.
Smallest Sequence of 5 Consecutive Numbers which are Happy
The smallest sequence of $5$ consecutive integers all of which are happy numbers is: :$44 \, 488, 44 \, 489, 44 \, 490, 44 \, 491, 44 \, 492$
Reciprocal of 21
The decimal expansion of the reciprocal of $21$ has period $6$: :$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$
Properties of 47,619
If you split $047 \, 619$ into two halves, they add up to $666$: :$047 + 619 = 666$ which is a multiple of $333$. and: : $047 \, 619 = 143 \times 333$ Similarly, you can split $047 \, 619$ into three thirds, and these add up to $99$: :$04 + 76 + 19 = 99$ and: : $047 \, 619 = 481 \times 99$ The square of $47 \, 619$: :$47 \, 619^2 = 2 \, 267 \, 569 \, 161$ can itself be split into two $6$-digit halves which together add to the recurring part of $\dfrac 4 7$: :$2267 + 569 \, 161 = 571 \, 428$ This is caused by the fact that $047 \, 619$ is the recurring part of the Reciprocal of 21: :$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$ where $21$ is the product of (the smallest) $2$ distinct primes which do not divide $10$. {{finish|for whatever value of "finish off" is determined}}
Smallest Fourth Power as Sum of 5 Fourth Powers
The smallest $4$th power which can be expressed as the sum of $5$ $4$th powers is: :$15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$
Boolean Interpretation is Well-Defined
Let $\LL_0$ be the language of propositional logic. Let $v: \LL_0 \to \set {\T, \F}$ be a boolean interpretation. Then $v$ is well-defined.
Pandigital Pairs whose Squares are Pandigital
The elements of the following pandigital pairs of integers each have squares which are themselves pandigital: :$\left({35 \, 172, 60 \, 984}\right), \left({57 \, 321, 60 \, 984}\right), \left({58 \, 413, 96 \, 702}\right), \left({59 \, 403, 76 \, 182}\right)$ {{OEIS|A085545|order = first}}
Language of Propositional Logic has Unique Parsability
The language of propositional logic $\mathcal L_0$ has unique parsability.
Powers of 2 not containing Digit Power of 2
$2^{16} = 65 \, 536$ is the only known power of $2$, up to $2^{31 \, 000}$, whose digits do not contain $1$, $2$, $4$ or $8$.
Construction of Regular 65,537-Gon
It is possible to construct a regular polygon with $65 \, 537$ sides) using a compass and straightedge construction.
There are no Odd Unitary Perfect Numbers
No unitary perfect numbers exist which are odd.
Automorphic Numbers with 5 Digits
The only $5$-digit automorphic number which does not begin with a zero is $90 \, 625$.
Kaprekar's Process on 5 Digit Number
Let $n$ be a $5$-digit integer whose digits are not all the same. Kaprekar's process, when applied to $n$, results in one of the following $3$ cycles: :$53 \, 955 \to 59 \, 994 \to 53 \, 955$ :$61 \, 974 \to 82 \, 962 \to 75 \, 933 \to 63 \, 954 \to 61 \, 974$ :$62 \, 964 \to 71 \, 973 \to 83 \, 952 \to 74 \, 943 \to 62 \, 964$
Tableau Extension Lemma/General Statement
Let $\mathbf H'$ be another finite set of WFFs. Then there exists a finished finite propositional tableau $T'$ such that: $(1):\quad$ the root of $T'$ is $\mathbf H \cup \mathbf H'$; $(2):\quad$ $T$ is a rooted subtree of $T'$.
Tableau Extension Lemma/General Statement/Proof 2
Let $T$ be a finite propositional tableau. Let its hypothesis set $\mathbf H$ be finite. {{:Tableau Extension Lemma/General Statement}}
Factors of Sums of Powers of 100,000
All integers $n$ of the form: :$n = \displaystyle \sum_{k \mathop = 0}^m 10^{5 k}$ for $m \in \Z_{> 0}$ are composite.
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
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 $\R$.
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
Recurring Part of Fraction times Period gives 9-Repdigit
Let a (strictly) positive integer $n$ be such that the decimal expansion of its reciprocal has a recurring part of period $d$. 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. {{mistake|Counterexample:1/6. Possibly true if n is not a multiple of 2 or 5}}
Properties of 142,857
This page gathers together some properties of $142 \, 857$ which arise through its being the digits of the recurring part of the reciprocal of $7$. Multiplication of $142 \, 857$ by numbers higher than $7$ produces a similar pattern to when you multiply it by a single digit, but with added complications. For example: {{begin-eqn}} {{eqn | l = 142 \, 857 \times 12       | r = 1 \, 714 \, 284       | c =  }} {{end-eqn}} This becomes $714 \, 285$ when you take the $1$ off the front and add it to the back. The exception is when you multiply it by $7$ or a multiple of $7$: {{begin-eqn}} {{eqn | l = 142 \, 857 \times 7       | r = 999 \, 999       | c =  }} {{eqn | l = 142 \, 857 \times 14       | r = 1 \, 999 \, 998       | c =  }} {{end-eqn}} From Recurring Part of Fraction times Period gives 9-Repdigit, it is seen that this property is shared of all numbers formed from the digits of the recurring part of a recurring fraction. If you divide $142 \, 857$ into two equal parts and add them, you get $999$: :$142 + 857 = 999$ Thus by Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999, $142 \, 857$ is divisible by $999$: :$142 \, 857 = 143 \times 999$ Also, we have: :$999 \, 999 = 1001 \times 999$ and so $999 \, 999$ is divisible by $999$. But as $999 \, 999 = 7 \times 142 \, 857$ we have that $999 \, 999$ is divisible by $7$. Thus it follows from Euclid's Lemma that $142 \, 857$ is divisible by $999$. {{finish|A lot of material from this chapter of Wells has been skipped, because it's just not very interesting. I leave it open for someone else to complete, if they want to.}}
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.
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 $999$  $n$ is divisible by $999$.
Number which is Sum of Subfactorials of Digits
The only integer which is the sum of the subfactorials of its digits is $148 \, 349$: :$148 \, 349 = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9$
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 $y$: :$24, 120, 175 \, 560$
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}}
Smallest Fifth Power which is Sum of 6 Fifth Powers
The smallest fifth power which is the sum of $6$ fifth powers is $12^5 = 248 \, 832$: :$12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$
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}}
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.
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).}}
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: :$4 \times 60 \times 65 = 8 \times 25 \times 78$
510,510 is Product of 4 Consecutive Fibonacci Numbers
$510 \, 510$ Can be expressed as the product of $4$ distinct Fibonacci numbers: :$510 \, 510 = 13 \times 21 \times 34 \times 55$ and is also the $7$th primorial: :$510 \, 510 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$
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 $\mathbf H'$.
Set of 7 Anagrams which are Square
The following integers are all anagrams, and all square: :$1 \, 048 \, 576, 1 \, 056 \, 784, 1 \, 085 \, 764, 5 \, 740 \, 816, 5 \, 764 \, 801, 6 \, 754 \, 801, 7 \, 845 \, 601$
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 in $X$ which is fixed by $g$, that is: :$X^g := \set {x \in X: g x = x}$ Then: :$\displaystyle \size {X / G} = \frac 1 {\order G} \sum_{g \mathop \in G} \size {X^g}$ In words, the number of orbits equals the average number of fixed elements.
Smallest Cunningham Chain of the First Kind of Length 7
The smallest Cunningham chain of the first kind of length $7$ is: :$\left({1 \, 122 \, 659, 2 \, 245 \, 319, 4 \, 490 \, 639, 8 \, 981 \, 279, 17 \, 962 \, 559, 35 \, 925 \, 119, 71 \, 850 \, 239}\right)$
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  $n$ is in the set: :$\set {\pm 15, \pm 22 \, 440, \pm 2 \, 759 \, 640}$
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!       | r = 6! \times 7!       | c =  }} {{end-eqn}}
Smallest Solution to Equation p^p times q^q = r^r
Consider the Diophantine equation: :$p^p \times q^q = r^r$ Its smallest solution is: {{begin-eqn}} {{eqn | l = p       | r = 12^6       | rr=  = 2 \, 985 \, 984 }} {{eqn | l = q       | r = 6^8       | rr=  = 1 \, 679 \, 616 }} {{eqn | l = r       | r = 2^{11} \times 3^7       | rr=  = 4 \, 478 \, 976 }} {{end-eqn}}
Existence of Matrix Logarithm
Let $T$ be a square matrix of order $n$. Then there exists a real matrix $S$ such that $e^S = T$ : :$(1): \quad  T$ is not a singular matrix and: :$(2): \quad $for every negative eigenvalue $\lambda$ of $T$ and for every positive integer $k$, the Jordan form of $T$ has an even number of $k \times k$ blocks associated with $\lambda$. pcu6qfvqi59x43t8w6ubk68j3p2rko5
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.
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.
Largest Integer Not Expressible as Sum of Distinct 4th Powers
The largest integer which cannot be expressed as the sum of distinct $4$th powers is $5 \, 134 \, 240$.
Properties of 12,345,679
$12 \, 345 \, 679$ has the following properties: {{begin-eqn}} {{eqn | l = 12 \, 345 \, 679 \times 1       | r = 12 \, 345 \, 679       | c = digit $8$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 2       | r = 24 \, 691 \, 358       | c = digit $7$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 3       | r = 37 \, 037 \, 037       | c =  }} {{eqn | l = 12 \, 345 \, 679 \times 4       | r = 49 \, 382 \, 716       | c = digit $5$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 5       | r = 61 \, 728 \, 395       | c = digit $4$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 6       | r = 74 \, 074 \, 074       | c =  }} {{eqn | l = 12 \, 345 \, 679 \times 7       | r = 86 \, 419 \, 753       | c = digit $2$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 8       | r = 98 \, 765 \, 432       | c = digit $1$ is missing }} {{eqn | l = 12 \, 345 \, 679 \times 9       | r = 111 \, 111 \, 111       | c =  }} {{end-eqn}} In each product, the sequence $1$ to $9$, with the one given digit missing, can be read in order by cycling round it, skipping a fixed number of digits (counting an extra one when going from start to end), for example: :$2 \ (4691) \ 3 \ (58?2) \ 4 \ (6913) \ 5 \ (8?24) \ 6 \ (9135) \ 8 (?246) \ 9$ {{expand|Add some mathematical analysis explaining this phenomenon}}
Number of Ways to Tile Standard Chessboard with Dominoes
The number of ways to tile a standard chessboard with dominoes is $12 \, 988 \, 816$
Sequence of 8 Consecutive Primes with Same Pattern of Differences as from 11
The $8$ prime numbers starting at $15 \, 760 \, 091$ have the same prime gaps as the $8$ primes starting at $11$. Those $8$ primes are: :$15 \, 760 \, 091, 15 \, 760 \, 093, 15 \, 760 \, 097, 15 \, 760 \, 099, 15 \, 760 \, 103, 15 \, 760 \, 109, 15 \, 760 \, 111, 15 \, 760 \, 117$ Their prime gaps are: :$2, 4, 2, 4, 6, 2, 6$ The $8$ primes starting at $11$: :$11, 13, 17, 19, 23, 29, 31, 37$ Their prime gaps are: :$2, 4, 2, 4, 6, 2, 6$
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 $33 \, 817 \, 088$.
Smallest Integer which is Sum of 2 Cubes in 4 Ways
The smallest positive integer which can be expressed as the sum of $2$ cubes in $4$ different ways is: {{begin-eqn}} {{eqn | l = 42 \, 549 \, 416       | r = 348^3 + 74^3       | c =  }} {{eqn | r = 282^3 + 272^3       | c =  }} {{eqn | r = \left({-2662}\right)^3 + 2664^3       | c =  }} {{eqn | r = \left({-475}\right)^3 + 531^3       | c =  }} {{end-eqn}}
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 subsequences:  Those where the consecutive integers are increasing: :$428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, \ldots$ {{OEIS|A030467}} Those where the consecutive integers are decreasing: :$91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, \ldots$ {{OEIS|A054216}}
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}}
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: :$83 \, 623 \, 935, 83 \, 623 \, 935 \times 83 \, 623 \, 937$
Hardy-Ramanujan Number/Examples/87,539,319
The $3$rd Hardy-Ramanujan number $\operatorname {Ta} \left({3}\right)$ is $87 \, 539 \, 319$: {{begin-eqn}} {{eqn | l = 87 \, 539 \, 319       | r = 167^3 + 436^3       | c =  }} {{eqn | r = 228^3 + 423^3       | c =  }} {{eqn | r = 255^3 + 414^3       | c =  }} {{end-eqn}}
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}}
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: :$\displaystyle \sum_{u \mathop \in S \mathop \cup T} f \left({u}\right) = \sum_{s \mathop \in S} f \left({s}\right) + \sum_{t \mathop \in T} f \left({t}\right)$
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: :$\displaystyle \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in S} \map f {\map \sigma s}$
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: :$\displaystyle \sum_{s \mathop \in S} f \left({s}\right) = \sum_{t \mathop \in T} f \left({g \left({t}\right)}\right)$
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 indexed summations: :$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{i \mathop = a}^b \map f {\map \sigma i}$
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: :$\displaystyle \sum_{i \mathop = a}^b f(i) = \sum_{i \mathop = a + c}^{b + c} \map f {i - c}$
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 $\left[{a \,.\,.\, c}\right]$ denote the integer interval between $a$ and $c$. Let $b \in \left[{a-1 \,.\,.\, c}\right]$. Let $f : \left[{a \,.\,.\, c}\right] \to \mathbb A$ be a mapping. Then we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^c f(i) = \sum_{i \mathop = a}^b f(i) + \sum_{i \mathop = b+1}^c f(i)$
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 $f : \{a, a+1\} \to \mathbb A$ be a real-valued function. Then the indexed summation: :$\displaystyle \sum_{i \mathop = a}^{a+1} f(i) = f(a) + f(a+1)$.
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 $f : \{a\} \to \mathbb A$ be a mapping on the singleton $\{a \}$. Then the indexed summation: :$\displaystyle \sum_{i \mathop = a}^{a} f(i) = f(a)$
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 indexed summations: :$\displaystyle \sum_{i \mathop = a}^b \map f i = \sum_{i \mathop = c}^d \map f {\map g i}$
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.
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 $a\leq b$. Let $\left[{a \,.\,.\, b}\right]$ be the integer interval between $a$ and $b$. Let $f : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a mapping. Then we have an equality of indexed summations: :$\displaystyle \sum_{i \mathop = a}^b f(i) = f(a) + \sum_{i \mathop = a+1}^b f(\sigma(i))$