User:Ascii/ProofWiki Sampling Notes for Theorems/Number Theory


 * 1) Set of Integers Bounded Below by Integer has Smallest Element
 * Let $\Z$ be the set of integers.
 * Let $\le$ be the ordering on the integers.
 * Let $\O \subset S \subseteq \Z$ such that $S$ is bounded below in $\struct {\Z, \le}$.
 * Then $S$ has a smallest element.
 * 1) Set of Integers Bounded Above by Integer has Greatest Element
 * Let $\Z$ be the set of integers.
 * Let $\le$ be the ordering on the integers.
 * Let $\O \subset S \subseteq \Z$ such that $S$ is bounded above in $\struct {\Z, \le}$.
 * Then $S$ has a greatest element.
 * 1) Division Theorem
 * $\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < \left\lvert{b}\right\rvert$
 * 1) Integer Divisor Results/One Divides all Integers
 * $\forall n \in \Z: 1 \divides n \land -1 \divides n$
 * 1) Integer Divisor Results/Integer Divides Itself
 * $\forall n \in \Z: n \divides n$
 * 1) Integer Divisor Results/Integer Divides its Negative
 * $\forall n \in \Z: n \divides (-n)$
 * 1) Integer Divisor Results/Integer Divides its Absolute Value
 * $\forall n \in \Z: n \divides \left \lvert {n}\right \rvert$
 * $\forall n \in \Z: \left \lvert {n}\right \rvert \divides n$
 * 1) Integer Divisor Results/Integer Divides Zero
 * $\forall n \in \Z: n \divides 0$
 * 1) Zero Divides Zero
 * $\forall n \in \Z: 0 \divides n \implies n = 0$
 * 1) Divisor Relation is Transitive
 * $\forall x, y, z \in \Z: x \divides y \land y \divides z \implies x \divides z$
 * 1) Divisor Relation is Antisymmetric
 * $\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$
 * 1) Divisor Relation on Positive Integers is Partial Ordering
 * The divisor relation is a partial ordering of $\Z_{>0}$.
 * 1) Integer Divisor Results/Divisors of Negative Values
 * $\forall m, n \in \Z: m \divides n \iff -m \divides n \iff m \divides -n \iff -m \divides -n$
 * 1) Common Divisor Divides Integer Combination
 * 2) Existence of Greatest Common Divisor
 * $\forall a, b \in \Z: a \ne 0 \lor b \ne 0$, there exists a largest $d \in \Z_{>0}$ such that $d \divides a$ and $d \divides b$.
 * That is, the greatest common divisor of $a$ and $b$ always exists.
 * 1) Greatest Common Divisor is at least 1
 * The greatest common divisor of $a$ and $b$ is at least $1$: $\forall a, b \in \Z_{\ne 0}: \gcd \set {a, b} \ge 1$
 * 1) GCD of Integer and Divisor
 * $a, b \in \Z_{>0}: a \divides b \implies \gcd \left\{{a, b}\right\} = a$
 * 1) GCD for Negative Integers
 * $\gcd \left\{{a, b}\right\} = \gcd \left\{{\left|{a}\right|, b}\right\} = \gcd \left\{{a, \left|{b}\right|}\right\} = \gcd \left\{{\left|{a}\right|, \left|{b}\right|}\right\}$
 * $\gcd \left\{{a, b}\right\} = \gcd \left\{{-a, b}\right\} = \gcd \left\{{a, -b}\right\} = \gcd \left\{{-a, -b}\right\}$
 * 1) GCD with Zero
 * $\forall a \in \Z_{\ne 0}: \gcd \left\{{a, 0}\right\} = \left\lvert{a}\right\rvert$
 * 1) Bézout's Lemma
 * Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
 * Then $\exists x, y \in \Z: a x + b y = \gcd \left\{{a, b}\right\}$
 * That is, $\gcd \left\{{a, b}\right\}$ is an integer combination (or linear combination) of $a$ and $b$.
 * Furthermore, $\gcd \left\{{a, b}\right\}$ is the smallest positive integer combination of $a$ and $b$.
 * 1) Solution of Linear Diophantine Equation
 * 2) GCD with Remainder
 * Let $a, b \in \Z$ and $q, r \in \Z$ such that $a = q b + r$.
 * Then $\gcd \left\{{a, b}\right\} = \gcd \left\{{b, r}\right\}$.
 * 1) Coprime Relation for Integers is Non-Reflexive
 * 2) Coprime Relation for Integers is Not Reflexive
 * 3) Coprime Relation for Integers is Not Antireflexive
 * 4) Coprime Relation for Integers is Symmetric
 * 5) Coprime Relation for Integers is Not Antisymmetric
 * 6) Coprime Relation for Integers is Non-transitive
 * 7) Coprime Relation for Integers is Not Transitive
 * 8) Coprime Relation for Integers is Not Antitransitive
 * 9) Integer Combination of Coprime Integers
 * Two integers are coprime there exists an integer combination of them equal to $1$: $\forall a, b \in \Z: a \perp b \iff \exists m, n \in \Z: m a + n b = 1$
 * 1) Divisor of Sum of Coprime Integers
 * Let $a, b, c \in \Z_{>0}$ such that: $a \perp b$ and $c \divides \paren {a + b}$
 * Then $a \perp c$ and $b \perp c$
 * 1) Integers Divided by GCD are Coprime
 * Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their GCD:
 * $\gcd \left\{{a, b}\right\} = d \iff \dfrac a d, \dfrac b d \in \Z \land \gcd \left\{{\dfrac a d, \dfrac b d}\right\} = 1$
 * 1) Euclid's Lemma
 * Let $a, b, c \in \Z$ and $a \divides b c$.
 * Then $a \divides c$.


 * 1) Existence of Lowest Common Multiple
 * Let $a, b \in \Z: a b \ne 0$.
 * The lowest common multiple of $a$ and $b$, denoted $\lcm \set {a, b}$, always exists.


 * 1) Product of GCD and LCM
 * $\lcm \set {a, b} \times \gcd \set {a, b} = \size {a b}$


 * 1) GCD and LCM Distribute Over Each Other
 * Let $a, b, c \in \Z$.
 * Then $\lcm \set {a, \gcd \set {b, c} } = \gcd \set {\lcm \set {a, b}, \lcm \set {a, c} }$
 * And $\gcd \set {a, \lcm \set {b, c} } = \lcm \set {\gcd \set {a, b}, \gcd \set {a, c} }$


 * 1) Divisor of One of Coprime Numbers is Coprime to Other

Possible:
 * 1) Divisors of Product of Coprime Integers
 * Let $a \divides b c$, where $b \perp c$.
 * Then $a = r s$, where $r \divides b$ and $s \divides c$.

= Congruence =


 * 1) Number is Divisor iff Modulo is Zero
 * 2) Integer is Congruent to Integer less than Modulus


 * 1) Congruence Modulo Integer is Equivalence Relation
 * For all $z \in \Z$, congruence modulo $z$ is an equivalence relation.
 * 1) Modulo Addition is Closed/Integers
 * Let $m \in \Z$ be an integer.
 * Then addition modulo $m$ on the set of integers modulo $m$ is closed: $\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m +_m \eqclass y m \in \Z_m$.
 * 1) Modulo Addition is Associative
 * $\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m +_m \eqclass y m} +_m \eqclass z m = \eqclass x m +_m \paren {\eqclass y m +_m \eqclass z m}$
 * 1) Modulo Addition is Commutative
 * Modulo addition is commutative: $\forall x, y, z \in \Z: x + y \pmod m = y + x \pmod m$
 * 1) Modulo Multiplication is Closed
 * Multiplication modulo $m$ is closed on the set of integers modulo $m$: $\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m \in \Z_m$.
 * 1) Modulo Multiplication is Associative
 * Multiplication modulo $m$ is associative: $\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m, \left[\!\left[{z}\right]\!\right]_m \in \Z_m: \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m}\right) \times_m \left[\!\left[{z}\right]\!\right]_m = \left[\!\left[{x}\right]\!\right]_m \times_m \left({\left[\!\left[{y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)$
 * 1) Modulo Multiplication is Commutative
 * Multiplication modulo $m$ is commutative: $\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m = \eqclass y m \times_m \eqclass x m$

The Binomial Theorem

 * 1) Binomial Theorem

Fibonacci

 * 1) Fibonacci Number with Negative Index
 * 2) Cassini's Identity
 * $F_{n + 1} F_{n - 1} - F_n^2 = \paren {-1}^n$
 * 1) Honsberger's Identity
 * 2) Fibonacci Number in terms of Larger Fibonacci Numbers


 * 1) Divisibility of Fibonacci Number
 * 2) Sum of Sequence of Fibonacci Numbers
 * 3) Sum of Sequence of Odd Index Fibonacci Numbers
 * 4) Sum of Sequence of Even Index Fibonacci Numbers
 * 5) Sum of Sequence of Products of Consecutive Fibonacci Numbers
 * 6) Lucas Number as Sum of Fibonacci Numbers


 * 1) Consecutive Fibonacci Numbers are Coprime
 * 2) GCD of Fibonacci Numbers
 * 3) Vajda's Identity/Formulation 1