# Euler pseudoprime  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] Code [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

A composite number n is called an Euler pseudoprime to a natural base a if $a^{\frac {n-1}{2}}\equiv 1{\pmod {n}}$ or $a^{\frac {n-1}{2}}\equiv \left(-1\right){\pmod {n}}$ ## Properties

$\left(a^{\frac {n-1}{2}}\right)^{2}=a^{n-1}$ and
$1^{2}=\left(-1\right)^{2}=1\$ • Every Euler Pseudoprime to base a that satisfies $a^{\frac {n-1}{2}}\equiv \left({\frac {a}{n}}\right){\pmod {n}}$ is an Euler-Jacobi pseudoprime.
• Strong pseudoprimes are Euler pseudoprimes too.

## Absolute Euler pseudoprime

An absolute Euler pseudoprime is a composite number c that satisfies the congruence $a^{\frac {c-1}{2}}\equiv 1{\pmod {c}}$ or $a^{\frac {n-1}{2}}\equiv \left(-1\right){\pmod {n}}$ for every base a that is coprime to c. Every absolute Euler pseudoprime is also a Carmichael number.