13

From Number
Jump to: navigation, search
This article is about a particular natural number.|View all articles on particular natural numbers

Summary

Factorization

The number 13 is a prime number.

Properties and families

Property or family Parameter values First few numbers Proof of satisfaction/membership/containment
prime number it is the 6th prime number 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, [SHOW MORE]View list on OEIS A natural number is prime if and only if is not divisible by any prime less than or equal to . Since is between 3 and 4, we only need to check divisibility by primes less than or equal to 3, i.e., we need to verify that 13 is not divisible by the primes 2 and 3.
Proth prime: prime of the form with 3, 5, 13, 17, 41, 97, 113, [SHOW MORE]View list on OEIS
regular prime fifth regular prime (2 is neither regular nor irregular) 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, [SHOW MORE]View list on OEIS

Polynomials

Prime-generating polynomials

Below are some polynomials that give prime numbers for small input values, which give the value 13 for suitable input choice.

Polynomial Degree Some values for which it generates primes Input value at which it generates 13
2 all numbers 1-10, because 11 is one of the lucky numbers of Euler. 2

Irreducible polynomials by Cohn's irreducibility criterion

By Cohn's irreducibility criterion, we know that if we write 13 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here all the irreducible polynomials:

Base 13 in base Corresponding irreducible polynomial
2 1101
3 111
4 31
5 23
6 21
7 16
8 15
9 14
10 13
11 12
12 11

Multiples

Interesting multiples

Number Prime factorization What's interesting about it
1105 5 times 13 times 17 second Carmichael number, i.e., absolute pseudoprime
1729 7 times 13 times 19 third Carmichael number, i.e., absolute pseudoprime
2821 7 times 13 times 31 fifth Carmichael number, i.e., absolute pseudoprime