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.

Further reading

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