Fermat pseudoprime

Builds on Composite number Congruence modulo n Fermat's little theorem

Definition

Let $a$ be an integer with $a > 1$ . A Fermat pseudoprime to the base $a$ is a composite number $n$ such that $a^{n-1} \equiv 1 \pmod n$ .

Remarks

Fermat's little theorem says that each prime $n$ with $n \nmid a$ satisfies $a^{n-1} \equiv 1 \pmod n$ . A Fermat pseudoprime is a composite number that passes the same test. The smallest Fermat pseudoprime to the base $2$ is $341 = 11 \cdot 31$ . The condition implies $\gcd(a, n) = 1$ . A composite number that is a Fermat pseudoprime to each base coprime to it is a Carmichael number.

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Divisibility definition
Congruence modulo n definition
Prime number definition
Fermat's little theorem statement
Composite number definition
Fermat pseudoprime definition

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Carmichael number

Tag	`02M2K4QH9P`
URI	`m2://02M2K4QH9P`
Kind	definition
Writing	W0 stub
MSC	11A51
Defines	Fermat pseudoprime
Uses	Composite number, Congruence modulo n, Fermat's little theorem

Fermat pseudoprime

Definition

Remarks

Reading path

Where this leads

Further reading

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