# Fermat pseudoprime

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A composite number  is called a Fermat pseudoprime to a natural base , which is coprime to , if .

## Restriction

It is sufficient that the base  satisfies  because every odd number  satisfies  for [1].

If  is a Fermat pseudoprime to base  then  is a Fermat pseudoprime to base  for every integer .

## Odd Fermat pseudoprimes

To every odd Fermat pseudoprime  exist an even number of bases . Every base  has a cobase .

Examples:

15 is a Fermat pseudoprime to the bases 4 and 11
49 is a Fermat pseudoprime to the bases 18, 19, 30 and 31

## Properties

Most of the pseudoprimes, like Euler pseudoprimes, Carmichael numbers, Fibonacci pseudoprimes and Lucas pseudoprimes, are Fermat pseudoprimes.

## References and notes

1. Richard E. Crandall and Carl Pomerance: Prime Numbers: A Computational Perspective. Springer-Verlag, 2001, page 132, Theorem 3.4.2.