# Category:Number Theory

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This category contains results about Number Theory.

Definitions specific to this category can be found in Definitions/Number Theory.

**Number theory** is the branch of mathematics which studies the properties of the natural numbers.

## Subcategories

This category has the following 200 subcategories, out of 241 total.

(previous page) (next page)### 0

- 0.999...=1 (6 P)

### A

- Ackermann Functions (2 P)
- Aliquot Sums (2 P)
- Amicable Triplets (4 P)
- Asymptotic Notation (24 P)
- Aurifeuillian Factorizations (2 P)

### B

- Beatty Sequences (10 P)
- Bernoulli Polynomials (3 P)
- Bertrand-Chebyshev Theorem (4 P)
- Bézout's Lemma (11 P)

### C

- Class Field Theory (empty)
- Collatz Conjecture (5 P)
- Computational Number Theory (empty)
- Continued Square Roots (empty)
- Cyclic Numbers (1 P)

### D

- Digital Roots (2 P)
- Dimensional Analysis (2 P)
- Diophantine m-Tuples (2 P)
- Dirichlet Convolution (8 P)
- Dismal Arithmetic (empty)
- Division Theorem (24 P)
- Double Factorials (1 P)

### E

- Elliptic Curves (1 P)

### F

- Farey Sequence (2 P)
- Fermat Quotients (2 P)
- Fermat's Last Theorem (3 P)
- Fermat's Little Theorem (13 P)
- Fleenor-Heronian Triangles (1 P)

### G

- Goldbach Conjecture (6 P)

### H

- Harmonic Mean (5 P)
- Harmonic Sequences (1 P)
- Hilbert-Waring Theorem (29 P)
- Hyperoperation (5 P)

### I

- Integer Sequences (2 P)
- Integral Ideals (4 P)
- Interesting Numbers (empty)

### K

- Kummer Theory (empty)

### L

- Legendre Symbol (9 P)
- Locker Problem (3 P)

### M

- Multinomial Coefficients (2 P)
- Möbius Function (8 P)

### N

- Normal Numbers (5 P)
- Number Fields (6 P)

### O

- Oesterlé-Masser Conjecture (5 P)
- Order Notation (4 P)

### P

- Pandigital Fractions (27 P)
- Pell Numbers (2 P)
- Penholodigital Integers (10 P)
- Permanents (1 P)
- Prime Gaps (10 P)
- Prime-Counting Function (6 P)
- Primitive Prime Factors (1 P)
- Product of Proper Divisors (1 P)

### Q

- Quadratic Irrationals (4 P)
- Quasiperfect Numbers (3 P)

### R

- Ramanujan-Nagell Equation (2 P)
- Rising Factorials (13 P)

### S

- Signum Function (5 P)

## Pages in category "Number Theory"

The following 179 pages are in this category, out of 179 total.

### 1

### A

### C

- Casting Out 9s
- Catalan's Conjecture
- Characteristic Function of Square-Free Integers is Multiplicative
- Chen's Theorem
- Chinese Remainder Theorem
- Collatz Conjecture
- Commensurability is Transitive
- Completely Additive Function is Additive
- Completely Multiplicative Function is Multiplicative
- Congruence Modulo 3 of Power of 2
- Congruence of Sum of Digits to Base Less 1
- Constant to Power of Number of Distinct Prime Divisors is Multiplicative Function
- Definition:Continued Square Root
- Continuum Property implies Well-Ordering Principle
- Coprimality Criterion
- Cube Root of 2 is Irrational
- Cyclic Permutations of 5-Digit Multiples of 41

### D

### E

- Eisenstein Integers form Integral Domain
- Eisenstein's Lemma
- Equality of Integers to the Power of Each Other
- Equivalence of Definitions of Second Chebyshev Function
- Euclidean Algorithm
- Euler's Pentagonal Numbers Theorem
- Euler's Product form of Riemann Zeta Function
- Euler's Sum of Powers Conjecture
- Euler's Theorem
- Euler's Theorem/Corollary 1
- Exact Form of Prime-Counting Function
- Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared
- Existence of Prime-Free Sequence of Natural Numbers
- Existence of q for which j - qk is Positive
- Existence of Rational Powers of Irrational Numbers

### F

### G

### H

### I

- Infinite Number of Primes of form 4n - 1
- Integer as Sum of Seven Positive Cubes
- Integer as Sum of Three Odd Squares
- Integer Less One divides Power Less One
- Integer Less One divides Power Less One/Corollary
- Integer Less One divides Power Less One/Corollary/Converse
- Integer to Power of Multiple of Order
- Integer to Power of Multiple of Order/Corollary
- Integer to Power of p-1 over 2 Modulo p
- Integer to Rational Power is Irrational iff not Integer or Reciprocal
- Integers as Sum of Three Pairwise Coprime Integers
- Integers Representable as Product of both 3 and 4 Consecutive Integers
- Integers with Primitive Roots
- Integral Multiple of an Algebraic Number
- Irrationality of Logarithm

### L

- Lagrange's Theorem (Number Theory)
- Largest Mutually Coprime Subset of Initial Segment of Natural Numbers
- Largest Prime Factor of n squared plus 1
- Length of Reciprocal of Product of Powers of 2 and 5
- Lifting The Exponent Lemma
- Liouville Numbers are Irrational
- Liouville's Theorem (Number Theory)
- Liouville's Theorem (Number Theory)/Corollary
- Local Maxima of Number of Goldbach Decompositions
- Locker Problem
- Lucas' Theorem
- Lucas' Theorem/Corollary

### N

- Natural Number is Divisor or Multiple of Divisor of Another
- Non-Zero Integer has Finite Number of Divisors
- Number of Bits for Decimal Integer
- Number of Digits in Number
- Number of Digits to Represent Integer in Given Number Base
- Number of Multiples less than Given Number
- Number Plus One divides Power Plus One iff Odd
- Numbers Divisible by Sum and Product of Digits
- Numbers Equal to Sum of Squares of Digits
- Numbers Not Expressible as Sum of no more than 5 Squares of Composite Numbers
- Numbers not Sum of Square and Prime
- Numbers whose Cyclic Permutations of 3-Digit Multiples are Multiples

### O

### P

- Positive Integer is Sum of Consecutive Positive Integers iff not Power of 2
- Positive Rational Number as Power of Number with Power of Itself
- Power Function is Completely Multiplicative/Integers
- Power of Sum Modulo Prime
- Power of Sum Modulo Prime/Corollary
- Prime Divisors of Cyclotomic Polynomials
- Prime Power of Sum Modulo Prime
- Prime Power of Sum Modulo Prime/Corollary
- Primes of form Power plus One
- Primitive Root is Generator of Reduced Residue System
- Principle of Mathematical Induction
- Product of Integers of form 4n + 1
- Product of Sums
- Product of Sums/Corollary
- Product of Three Consecutive Integers is never Perfect Power

### R

- Ramanujan's Infinite Nested Roots
- Rational Number as Power of Number with Itself
- Rational Square Root of Integer is Integer
- Real Logarithm is Completely Additive
- Recurring Parts of Multiples of One Fifty-Third
- Recurring Parts of Multiples of One Thirteenth
- Richert's Theorem
- Riemann Hypothesis implies Odd Number is Sum of at most 5 Primes
- Riemann Zeta Function and Prime Counting Function

### S

- Set of Divisors of Integer
- Set of Integers Bounded Above by Integer has Greatest Element
- Set of Integers Bounded Above has Greatest Element
- Set of Integers Bounded Below has Smallest Element
- Set of Successive Numbers contains Unique Multiple
- Sets of 4 Integers a, b, c, d for which Every Integer is in form ax^2 + by^2 + cz^2 + du^2
- Signum Complement Function on Natural Numbers as Characteristic Function
- Signum Function on Integers is Extension of Signum on Natural Numbers
- Signum Function on Natural Numbers as Characteristic Function
- Smallest n for which 2^n-3 is Divisible by n
- Smallest Positive Integer not of form +-4 mod 9 not representable as Sum of Three Cubes
- Square of Quadratic Gauss Sum
- Squares Ending in 5 Occurrences of 2-Digit Pattern
- Squares Ending in n Occurrences of m-Digit Pattern
- Squares Ending in n Occurrences of m-Digit Pattern/Example
- Structure of Recurring Decimal
- Sum of Odd Positive Powers
- Sum of Reciprocals of Divisors equals Abundancy Index
- Sum of Reciprocals of Primes is Divergent/Lemma
- Sum of two Fourth Powers cannot be Fourth Power
- Sum Over Divisors Equals Sum Over Quotients