07-03-2026, 07:25 PM
Fermat pseudoprime
Summary
A Fermat pseudoprime is a composite number that “pretends” to be prime by satisfying the same condition that true primes obey in Fermat’s Little Theorem: for a chosen base (a), it can happen that ($a^{n-1} \equiv 1 \pmod{n}$) even though (n) is not actually prime. This makes it a kind of mathematical imposter that fools a basic primality test. The concept matters because it shows why Fermat’s test alone is unreliable—some composite numbers slip through undetected, which is a big deal in number theory and cryptography.
A stronger and even rarer case is when a composite number satisfies the condition for every base coprime to it; these are called Carmichael numbers and are especially good at deceiving simple tests. Overall, Fermat pseudoprimes highlight the gap between “looks prime” and “is truly prime,” motivating more robust primality-testing methods used in modern mathematics and computing.
ARTICLE
Summary
A Fermat pseudoprime is a composite number that “pretends” to be prime by satisfying the same condition that true primes obey in Fermat’s Little Theorem: for a chosen base (a), it can happen that ($a^{n-1} \equiv 1 \pmod{n}$) even though (n) is not actually prime. This makes it a kind of mathematical imposter that fools a basic primality test. The concept matters because it shows why Fermat’s test alone is unreliable—some composite numbers slip through undetected, which is a big deal in number theory and cryptography.
A stronger and even rarer case is when a composite number satisfies the condition for every base coprime to it; these are called Carmichael numbers and are especially good at deceiving simple tests. Overall, Fermat pseudoprimes highlight the gap between “looks prime” and “is truly prime,” motivating more robust primality-testing methods used in modern mathematics and computing.
ARTICLE
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