Exercise 11, § 2

a) Let p be a prime number $\neq 2$. Then the following properties are equivalent:

$\alpha)$ $b_n/n \not\equiv 0$ (mod. p) for every even n such that $p - 1$ does not divide n.

$\beta)$ $b_n \not\equiv 0$ (mod. p) for $n = 2, 4, 6, \ldots, p - 3$. (Use exerc. 9b).)

b) One says that p is a regular prime number if it is equal to 2 or satisfies the equivalent conditions of a); otherwise p is called irregular. The prime numbers 2, 3, 5, 7, 11 are regular; 691 is irregular. Let I be a finite set of irregular primes. Let n be an integer $\geqslant 2$ multiple of $\prod_{q \in I} (q - 1)$ and such that $|b_n/n| > 1$, (cf. VI, p. 288, formula (15)). Let p be a prime factor of the numerator of $|b_n/n|$ in irreducible form. Show that $p - 1$ does not divide n and that consequently p is irregular. Deduce that the set of irregular prime numbers is infinite.