Category:Number Theory

A

• [+] Algebraic Number Theory
• [+] Arithmetic

C

• [+] Continued Fractions

D

• [+] Diophantine Equations
• [+] Divisibility Tests

E

• [+] Euclid Book IX

F

• [+] Factorials
• [+] Fibonacci Numbers
• [+] Figurate Numbers
• [+] Floor and Ceiling

G

• [+] Generating Functions

I

• [+] Interesting Numbers
• [+] Irrationality Proofs

M

• [+] Modulo Arithmetic

P

• [+] Prime Numbers
• [+] Pythagorean Triples

Q

• [+] Quadratic Residues

S

• [+] Sums of Sequences
• [+] Sums of Squares

0

• 0.999...=1

1

• 1+2+...+n+(n-1)+...+1 = n^2

A

• Absolute Value Bounded Below by Zero

B

• Backwards Induction
• Basic Properties of Multiplicative Function
• Basis Representation Theorem
• Bezout's Lemma
• Bounds of GCD for Sum and Difference Congruent Squares
• Bézout's Identity

C

• Chen's Theorem
• Common Divisor Divides Integer Combination
• Composite Number has Two Divisors Less Than It
• Congruence of Sum of Digits to Base Less 1
• Consecutive Integers are Coprime

D

• Divide by GCD for Coprime Integers
• Divides is Partial Ordering on Positive Integers
• Division Theorem
• Divisor of Sum of Coprime Integers
• Divisors of Product of Coprime Integers

E

• Equivalence of Definitions of the Second Chebyshev Function
• Equivalence of Riemann Zeta Function Definitions
• Euclid's Lemma
• Euclid's Lemma for Prime Divisors
• Euclidean Algorithm
• Euler Phi Function Even for Argument Greater than 2
• Euler Phi Function is Multiplicative
• Euler Phi Function of Product with Prime
• Euler Phi Function of a Prime
• Euler Phi Function of an Integer
• Euler's Criterion
• Euler's Theorem
• Existence of Greatest Common Divisor
• Existence of Lowest Common Multiple

F

• Factors of Mersenne Numbers
• Fermat's Last Theorem
• Fermat's Little Theorem (with corollary)
• First Supplement to the Law of Quadratic Reciprocity

G

• GCD and LCM Distribute Over Each Other
• GCD and LCM from Prime Decomposition
• GCD at least 1
• GCD for Negative Integers
• GCD iff Divisible by Common Divisor
• GCD is Smallest Positive Integer Combination
• GCD of Integer and Divisor
• GCD with Prime
• GCD with Remainder
• GCD with Zero
• Gauss's Lemma (Number Theory)

I

• Identity Function is Completely Multiplicative
• Ingham's Theorem on Convergent Dirichlet Series
• Integer Absolute Value Greater than Divisors
• Integer Combination of Coprime Integers
• Integer Divisor Results
• Integer has Order Modulo n iff Coprime to n
• Integer to Power of Multiple of Order
• Integers Bounded Above has Maximal Element
• Integers Bounded Below has Minimal Element
• Integers with Primitive Roots

L

• Law of Quadratic Reciprocity
• Lucas' Theorem
• Lucas-Lehmer Test

M

• Moebius Function is Multiplicative

N

• Not Coprime means Common Prime Factor

O

• Odd Integer 2n + 1
• Order Modulo n of Power of Integer
• Order of the Moebius Function

P

• Parity of Integer equals Parity of its Square
• Prime Power of Sum Modulo Prime
• Primitive Root is Generator of Reduced Residue System
• Principle of Mathematical Induction
• Product Form of Sum on Completely Multiplicative Function
• Product of Coprime Factors
• Product of GCD and LCM
• Product of Sigma and Euler Phi Functions
• Products of Sums

P cont.

• Properties of Legendre Symbol

Q

• Quadratic Nature by Sum of Floors of Multiples of Fractions
• Quotient and Remainder to Number Base

S

• Second Principle of Finite Induction
• Second Principle of Mathematical Induction
• Second Supplement to the Law of Quadratic Reciprocity
• Set of Divisors of an Integer
• Sigma Function Odd Iff Argument is Square or Twice Square
• Sigma Function is Multiplicative
• Sigma of Power of Prime
• Sigma of Prime Number
• Sigma of an Integer
• Square Root of 2 Is Irrational
• Squares with All Odd Digits
• Sum Over Divisors Equals Sum Over Quotients
• Sum Over Divisors of Multiplicative Function
• Sum of Euler Phi Function over Divisors
• Sum of Moebius Function over Divisors
• Sum of Odd Positive Powers
• Sum of Reciprocals of Divisors equals Abundancy
• Sum of Sequence of Squares

T

• Tau Function - Number of Positive Divisors
• Tau Function Odd Iff Argument is Square
• Tau Function is Multiplicative
• Tau Function of Square-Free Integer is Power of 2
• Tau of Power of Prime
• The sum of two even integers is itself an even number
• Theorem of Even Perfect Numbers
• There exist irrational a and b such that a^b is rational

U

• Unity Function is Completely Multiplicative
• Upper Bounds for Prime Numbers

V

• Values of Legendre Symbol

W

• Waring's Problem
• Well-Ordering Principle

Z

• Zero Divides Zero
• Zero and One are the only Consecutive Perfect Squares