Exercise 11, § 2

a) Let $p$ be a prime number $\neq 2$. Then the following properties are equivalent: α) $b_n/n \not\equiv 0 \pmod{p}$ for every even $n$ such that $p - 1$ does not divide $n$. β) $b_n \not\equiv 0 \pmod{p}$ for $n = 2, 4, 6, \ldots, p - 3$. (Use Exercise 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 said to be irregular. The prime numbers 2, 3, 5, 7, 11 are regular; 691 is irregular. Let $I$ be a finite set of irregular prime numbers. Let $n$ be an integer $\geq 2$ multiple of $\prod_{q \in I} (q - 1)$ and such that $|b_n/n| > 1$, (cf. VI, p. 19, formula (15)). Let $p$ be a prime factor of the numerator of $|b_n/n|$ put in irreducible form. Prove that $p - 1$ does not divide $n$ and consequently that $p$ is irregular. Deduce that the set of irregular prime numbers is infinite.