By Euler L.

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We say “presumably” because little is known in the way of rigorous density bounds, yet empirical evidence and heuristic arguments suggest relative rarity. For any odd prime p, Fermat’s “little theorem” tells us that 2p−1 ≡ 1 (mod p). 14) such primes being called Wieferich primes. These special primes ﬁgure strongly in the so-called ﬁrst case of Fermat’s “last theorem,” as follows. 14). Equivalently, we say that p is a Wieferich prime if the Fermat quotient qp (2) = 2p−1 − 1 p vanishes (mod p). One might guess that the “probability” that qp (2) so vanishes is about 1/p.

Prime-producing formulae are often amusing but, relatively speaking, useless. There is a famous counterexample though. 4). 3 Chapter 1 PRIMES! Primes of special form By prime numbers of special form we mean primes p enjoying some interesting, often elegant, algebraic classiﬁcation. For example, the Mersenne numbers Mq and the Fermat numbers Fn deﬁned by Mq = 2q − 1, n Fn = 22 + 1 are sometimes prime. These numbers are interesting for themselves and for their history, and their study has been a great impetus for the development of computational number theory.

4. However, we should also compare the chance of Mq being prime with a random number of the same size. 2 already indicates. Let us ignore for a moment the intricacies of this theorem and use only that Mq has no prime factors in the interval [1, q]. Here q is about lg Mq (here and throughout the book, lg means log2 ). What is the chance that a random number near x whose least prime factor exceeds lg x is prime? We know how to answer this question rigorously. First consider the chance that a random number near x has its least prime factor exceeding lg x.