6 Infinitely many irregular primes

This chapter proves Carlitz’s theorem [ 1 ] that there are infinitely many irregular primes (definition 1.1); the infinitude was first established by Jensen [ 3 ] . The strategy is a finite-set escape: given any finite set \(S\) of primes, one manufactures an even index \(M\), divisible by \(q-1\) for every prime \(q \in S\), such that \(|B_M/M| {\gt} 1\). Any prime \(p\) dividing the numerator of \(B_M/M\) then satisfies \((p-1) \nmid M\) by von Staudt–Clausen, which simultaneously rules \(p\) out of \(S\) and — through the unrestricted Kummer congruence, which moves the divisibility into Kummer’s range — proves that \(p\) is irregular.