A composite integer n such that for every integer a coprime to n, a^{n−1} ≡ 1 (mod n); therefore n behaves like a prime in the Fermat congruence for all bases coprime to it.

Principle

Captures composite moduli that pass the Fermat primality test for every base coprime to them; equivalently characterized by Korselt-like conditions (squarefree and each prime divisor p satisfies p−1 | n−1).

Demonstration

The smallest example is 561 = 3·11·17; for any a coprime to 561, a^{560} ≡ 1 (mod 561), so 561 is a Carmichael number and would consistently fool a simple Fermat test.

Misapplication

Referring to any Fermat pseudoprime (which may hold the congruence for some bases but not all) as a Carmichael number, or assuming every number satisfying a^{n−1} ≡ 1 for many small bases is Carmichael without checking the condition for all coprime a.

Consequence

Shows limitations of naive primality tests: the existence of Carmichael numbers requires stronger deterministic or probabilistic tests; their distribution affects reliability thresholds for base-based tests.

Reversal

Primes are the reversal: they satisfy the same Fermat congruence but are truly prime; another reversal is composite numbers that fail the congruence for some coprime bases and thus are easily distinguished by Fermat tests.

Boundary

Defined only for composite integers; requires the congruence to hold for every integer coprime to n. Excludes Fermat pseudoprimes that satisfy the congruence only for a subset of bases and excludes prime numbers.

Semantic Tension

Competes conceptually with Fermat pseudoprime, Euler pseudoprime and strong pseudoprime notions; the tension is between universal base-based deception (Carmichael) and deception for particular bases (pseudoprimes).

Synthesis

A Carmichael number is a composite, squarefree integer whose multiplicative structure forces the Fermat congruence to hold for every coprime base: a universal, base-independent pseudoprime that exposes the limits of simple primality checks.