Reference

De Morgan's Law
For statement variables and , we have

For statement variables , , and , the following rules hold for the logical operators and

Transitivity of Divisibility (TD)
For all integers , and , if and , then .

Divisibility of Integer Combinations (DIC)
For all integers , and , if and , then for all integers and , .

Pascal's Identity (PI)
For all positive integers and with , we have

Binomial Theorem 1 (BT 1)
For all integers and for all real numbers ,

Binomial Theorem 2 (BT 2)
For all integers and for all real numbers and ,

Bounds by Divisibility (BBD)
For all integers and , if and then

Division Algorithm (DA)
For all integers and positive integers , there exist unique integers and such that
and .

GCD with Remainder (GCD WR)
For all integers , , and , if , then

GCD Characterization Theorem (GCD CT)
For all integers and , and non-negative integers , if is a common divisor of and , and there
exist integers and such that , then

Bézout's lemma (BL)
For all integers and , there exist integers and such that , where .

Common Divisor Divides GCD (CDD GCD)
For all integers , and , if and then .

Coprimeness Characterization Theorem (CCT)
For all integers and , if and only if there exists integers and such that .

Division by the GCD (DB GCD)
For all integers and , not both zero, , where .

Coprimeness and Divisibility (CAD)
For all integers , and , if and , then .

Prime Factorization (PF)
Every natural number can be written as a product of primes

Euclid's Theorem (ET)
The number of primes is infinite.

Euclid's Lemma (EL)
For all integers and , and prime numbers , if , then or

unique factorization theorem (UFT)
Every natural number can be written as a product of primes uniquely, apart from the order of the factors

Divisors From Prime Factorization (DFPF)
Let and be positive integers, and let be a way to represent as a product of the distinct primes , where some or all of the exponents may be zero. The integers is a positive divisor of if and only if can be represented as a product , where for .

GCD From Prime Factorization (GCD PF)
Let and be positive integers, and let , and , be ways to express and as products of the distinct primes where some or all of the exponents may be zero. Then where for .

Linear Diophantine Equation Theorem, Part 1 (LDET 1)
For all integers , , and , with and not both zero, the linear Diophantine equation (in variables and ) has an integer solution if and only if , where

Linear Diophantine Equation Theorem, Part 2 (LDET 2)
Let , , and be integers with and both not zero, and define . If and is one particular integer solution to the linear Diophantine equation , then the set of all solutions is given by

Congruence Add and Multiply (CAM)
For all positive integers , if for all , then , and .

Congruence Power (CP)
For all positive integers and integers and , if , then .

Congruence Divide (CD)
For all integers , , and , if , and , then .

congruent iff same remainder (CISR)
For all integers and , if and only if and have the same remainder when divided by .

congruent to remainder (CTR)
For all integers and with , if and only if has remainder when divided by .

linear congurence theorem (LCT)
For all integers and , with non-zero, the linear congruence has a solution if and only if , where . Moreover, if is one particular solution to this congruence, then the set of all solutions is given by , or, equivalently, .

Modular Arithmetic Theorem (MAT)
For all integers and , with non-zero, the equation in has a solution if and only if , where . Moreover, when , there are $d solutions, given by , where is one particular solution.

Inverses in Z_m (INV Z_m)
Let be an integer with . The element in has a multiplicative inverse if and only if . Moreover, when , the multiplicative inverse is unique.

Inverses in Z_p (INV Z_p)
For all prime numbers and non-zero elements in , the multiplicative inverse exists and is unique.

Fermat's Little Theorem (FℓT)
For all prime numbers and integers not divisible by , we have

Corollary of Fermat's Little Theorem (Corollary of FℓT)
For all prime numbers and integers , we have

Chinese Remainder Theorem (CRT)
For all integers and , and positive integers and , if , then the simultaneous linear congruences

have unique solution modulo . Thus, if is one particular solution, then the solutions are given by the set of all integers such that .

Splitting Modulus Theorem (SMT)
For all integers and positive integers and , if , then the simultaneous congruences

have exactly the same solutions as the single congruence .

RSA Works (RSA)
For all integers , and , if

1. and are distinct primes
3. and are positive integers such that and
5. where
6. where

then

Properties of Conjugate (PCJ)
For the complex conjugate, the following properties hold for all :

3. and
5. If , then

Triangle Inequality (TIQ)
For all , we have

Properties of Modulus (PM)
For the modulus, the following properties hold for all :

1. if and only if
5. If , then

Polar Multiplication in C (PMC)
For all complex numbers and , we have .

De Moivre's Theorem (DMT)
For all real numbers and integers , we have .

Complex n-th Roots Theorem (CNRT)
For all complex numbers and natural numbers , the complex n-th roots of are given by .

Quadratic Formula (QF)
For all complex numbers , , and, with , the solutions to are given by , where is a solution to

Degree of a Product (DP)
For all fiends , and all non-zero polynomials and in , we have

Division Algorithm for Polynomials (DAP)
For all fields , and all polynomials and in with not the zero polynomial, there exist unique polynomials and in such that , where is the zero polynomial, or

Remainder Theorem (RT)
For all fiends , all polynomials , and all , the remainder polynomial when is divided by is the constant polynomial .

Factor Theorem (FT)
For all fields and all polynomials , and all , the linear polynomial is a factor of the polynomial if and only if (equivalently, is a root of the polynomial )

Fundamental Theorem of Algebra (FTA)
For all complex polynomials with , there exists a such that

Complex Polynomials of Degree n Have n Roots (CPN)
For all integers , and all complex polynomials of degree , there exists complex numbers and such that . Moreover, the roots of are .

Conjugate Roots Theorem (CJRT)
For all polynomials with real coefficients, if is a root of , then is a root of .

Real Quadratic Factors (RQF)
For all polynomials with real coefficients, if is a root of , and , then there exists a real quadratic polynomial and a real polynomial such that . Moreover, the quadratic factor is irreducible in .

Real Factors of Real Polynomials (RFRP)
For all real polynomials of positive degree, can be written as a product of real linear and real quadratic factors.