Pseudoprime

A pseudoprime is a composite number that has certain properties in common with prime numbers.

Introduction

If you want to find out if a given number is a prime number, you can to test it based on some properties that all prime numbers share. A property of prime numbers is that they are only divisible by one and itself. This is a defining property: it holds for all prime numbers and no other numbers.

However, other properties hold for all prime numbers and also some other numbers. For instance, every prime number, greater than 3, has the form $6n-1\$ or $6n+1\$ (with n an integer), but there are also composite numbers of this form: 25, 35, 49, 55, 65, 77, 85, 91, … . So, you could say that 25, 35, 49, 55, 65, 77, 85, 91, … are pseudoprimes with respect to the property of being of the form $6n-1\$ or $6n+1\$. There exist better properties, which lead to special pseudoprimes:

Different kinds of pseudoprimes

Property	kind of pseudoprime
$a^{n-1}\equiv 1{\pmod {n}}$	Fermat pseudoprime
$a^{\frac {n-1}{2}}\equiv 1{\pmod {n}}$	Euler pseudoprime
$a^{\frac {n-1}{2}}\equiv (n-1){\pmod {n}}$
$a^{d}\equiv 1{\pmod {n}}$	strong pseudoprime
$a^{d\cdot 2^{r}}\equiv -1{\pmod {n}}$
$a^{n}-a\$ is divisible by $n\$	Carmichael number
$P_{n}\$ is divisible by $n\$	Perrin pseudoprime
$V_{n}(P,Q)-P\$ is divisible by $n\$

Table of smallest Pseudoprimes

smallest Pseudoprimes
Number	Kind of Pseudoprime	Bases
15	Fermat pseudoprime	4, 15
21	Euler pseudoprime	8, 13
49	strong pseudoprime	18, 19, 30, 31
561	Carmichael number
1729	absolute Euler pseudoprime