Posted on Thursday, December 02, 2004 - 6:41 pm:

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A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any . In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable." As more simply noted by Derbyshire (2004, p. 33), "On balance, 2 pays its way [as a prime] on balance; 1 doesn't."

With 1 excluded, the smallest prime is therefore 2. However, since 2 is the only even prime, it is also somewhat special, and the set of all primes excluding 2 is therefore called the "odd primes." Note also that while 2 is considered a prime today, at one time it was not (Tietze 1965, p. 18; Tropfke 1921, p. 96).

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