Pseudoprime

pseudoprime, a composite, or nonprime, number n that fulfills a mathematical condition that most other composite numbers fail. The best-known of these numbers are the Fermat pseudoprimes. In 1640 French mathematician Pierre de Fermat first asserted “Fermat’s Little Theorem,” also known as Fermat’s primality test, which states that for any prime number p and any integer a such that p does not divide a (in this case, the pair are called relatively prime), p divides exactly into a^p − a. Although a number n that does not divide exactly into aⁿ − a for some a must be a composite number, the converse (that a number n that divides evenly into aⁿ − a must be prime) is not necessarily true. For example, let a = 2 and n = 341, then a and n are relatively prime and 341 divides exactly into 2³⁴¹ − 2. However, 341 = 11 × 31, so it is a composite number. Thus, 341 is a Fermat pseudoprime to the base 2 (and is the smallest Fermat pseudoprime). Thus, Fermat’s primality test is a necessary but not sufficient test for primality. As with many of Fermat’s theorems, no proof by him is known to exist. The first known proof of this theorem was published by Swiss mathematician Leonhard Euler in 1749.

There exist some numbers, such as 561 and 1,729, that are Fermat pseudoprime to any base with which they are relatively prime. These are known as Carmichael numbers after their discovery in 1909 by American mathematician Robert D. Carmichael.