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Carmichael number - mklabgr - 06-27-2026
Carmichael number
Summary
Carmichael numbers are one of the most fascinating surprises in number theory: they are composite numbers that behave like prime numbers in certain mathematical tests. Fermat’s Little Theorem says that prime numbers have the property that $(a^{p-1}\equiv1 \pmod p)$, and Carmichael numbers are special because they satisfy a similar rule for every number that is relatively prime to them, even though they are not actually prime.
Because of this, they are known as Fermat pseudoprimes or “prime impostors” and can fool simple primality tests used in computing. The smallest example is $561 = 3 × 11 × 17$, which looks prime under Fermat’s test but is actually composite. These mysterious numbers show how mathematics can have unexpected exceptions and why stronger methods are needed when checking whether very large numbers are truly prime.
ARTICLE