# Category:Prime Numbers

This category contains results about Prime Numbers.

Definitions specific to this category can be found in Definitions/Prime Numbers.

A **prime number** $p$ is a positive integer that has exactly two divisors which are themselves positive integers.

## Subcategories

This category has the following 60 subcategories, out of 60 total.

### B

### C

### E

### F

### G

### I

### L

### M

### N

### O

### P

### R

### S

### T

### U

### W

## Pages in category "Prime Numbers"

The following 156 pages are in this category, out of 156 total.

### A

### C

### D

### E

- Equal Consecutive Prime Number Gaps are Multiples of Six
- Equivalence of Definitions of Prime Number
- Euclid's Lemma for Prime Divisors
- Euclid's Theorem
- Euler Phi Function of Prime
- Euler Phi Function of Prime Power
- Euler Phi Function of Prime Power/Corollary
- Euler's Theorem/Corollary 1
- Existence of Prime between Prime and Factorial
- Existence of Prime-Free Sequence of Natural Numbers
- Existence of Product of Three Distinct Primes between n and 2n
- Exponents of Primes in Prime Decomposition are Less iff Divisor

### F

### G

### I

- Infinite Number of Chen Primes
- Infinite Number of Primes of form 4n - 1
- Integer as Sum of 27 Primes
- Integer Greater than 121 is Sum of Distinct Primes of form 4 n + 1
- Integer Greater than 205 is Sum of Distinct Primes of form 6 n + 1
- Integer is Expressible as Product of Primes
- Integers whose Number of Representations as Sum of Two Primes is Maximum
- Integers with Prime Values of Sigma Function
- Interval containing Prime Number of forms 4n - 1, 4n + 1, 6n - 1, 6n + 1

### L

### N

- Natural Number is Prime or has Prime Factor
- No Arithmetic Progression of 4 Primes with Common Difference 2
- No Arithmetic Progression of 4 Primes with Common Difference 2/Corollary
- Not Coprime means Common Prime Factor
- Number as Sum of Distinct Primes greater than 11
- Number of Quadratic Residues of Prime
- Number of Representations as Sum of Two Primes
- Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value
- Numbers not Sum of Square and Prime
- Numbers of Primes with at most n Digits
- Numbers that cannot be made Prime by changing 1 Digit
- Numbers with 6 or more Prime Factors
- Numbers with 7 or more Prime Factors

### O

### P

- Palindromic Primes in Base 10 and Base 2
- Positive Integer Greater than 1 has Prime Divisor
- Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes
- Positive Integers which Divide Sum of All Lesser Primes
- Positive Integers which Divide Sum of All Lesser Primes/Examples
- Power of Sum Mod Prime
- Power of Sum Mod Prime/Corollary
- Prime between n and 9 n divided by 8
- Prime Decomposition of 2^58+1
- Prime Decomposition of Integer is Unique
- Prime Divides Power
- Prime Equal to Sum of Digits of Cube
- Prime equals Plus or Minus One modulo 6
- Prime iff Coprime to all Smaller Positive Integers
- Prime iff Equal to Product
- Prime not Divisor implies Coprime
- Prime Number Formed by Concatenating Consecutive Integers down to 1
- Prime Number has 4 Integral Divisors
- Prime Number is Deficient
- Prime Number Theorem
- Prime Numbers Composed of Strings of Consecutive Ascending Digits
- Prime Numbers Embedded in Digits of Pi
- Prime Numbers of form Factorial Minus 1
- Prime Numbers of form Factorial Plus 1
- Prime Power of Sum Modulo Prime
- Prime Power of Sum Modulo Prime/Corollary
- Prime to Own Power minus 1 over Prime minus 1 being Prime
- Prime Triplet is Unique
- Prime-Generating Quadratics of form 2 a squared plus p
- Primes Expressible as x^2 + n y^2 for all n from 1 to 10
- Primes for which Powers to Themselves minus 1 have Common Factors
- Primes not Difference between Powers of 2 and 3
- Primes of Form n^2 + 1
- Primes of form Power Less One
- Primes of form Power of Two plus One
- Primes which are Mean of Neighboring Primes
- Primes whose Digits are Consecutive Ascending from 1

### R

### S

- Sequence of 11 Primes by Trebling and Adding 16
- Sequence of 5 Consecutive Non-Primable Numbers by Changing 1 Digit
- Sequence of 7 Consecutive Integers including Multiple of Prime greater than 41
- Sequence of 9 Primes of form 4n+1
- Sequence of Integers whose Factorial minus 1 is Prime
- Sequence of Integers whose Factorial plus 1 is Prime
- Sequence of Numbers Divisible by Sequence of Primes
- Set of Rational Numbers whose Numerator Divisible by p is Closed under Addition
- Set of Rational Numbers whose Numerator Divisible by p is Closed under Multiplication
- Sets of 4 Prime Quadruples
- Sieve of Eratosthenes
- Sigma Function of Power of Prime
- Sigma Function of Prime Number
- Smallest 10 Primes in Arithmetic Progression
- Smallest 17 Primes in Arithmetic Progression
- Smallest 18 Primes in Arithmetic Progression
- Smallest 22 Primes in Arithmetic Progression
- Smallest 5 Consecutive Primes in Arithmetic Progression
- Smallest n such that 6 n + 1 and 6 n - 1 are both Composite
- Smallest Odd Number not of form 2 a squared plus p
- Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways
- Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways
- Smallest Prime Expressible as x^2 + n y^2 for n from 1 to 10
- Smallest Titanic Palindromic Prime
- Solutions to p^2 Divides 10^p - 10
- Square Root of Prime is Irrational
- Sum of 714 and 715
- Sum of Distinct Primes of form 6n-1
- Sum of Pandigital Triplet of 3-Digit Primes
- Sum of Reciprocals of Twin Primes
- Sum of Sequence of Alternating Positive and Negative Factorials being Prime
- Sum of Sequence of Squares of Primes
- Sum over k to p over 2 of Floor of 2kq over p