Irregular Prime Divisors of the Bernoulli Numbers

If p is an irregular prime, <img width="82" height="37" align="MIDDLE" border="0" src="images/img4.gif" alt="$ p < 8000$">, then the indices 2n for which the Bernoulli quotients <!-- MATH ${B_{2n}}/2n$ --> are divisible by are completely characterized. In particular, it is always true that p$"> and that <!-- MATH ${B_{2n}}/2n\;\nequiv({B_{2n + p - 1}}/2n + p - 1)\pmod {p^2}$ --> if (p,2n) is an irregular pair. As a result, we obtain another verification that the cyclotomic invariants of Iwasawa all vanish for primes <img width="82" height="37" align="MIDDLE" border="0" src="images/img10.gif" alt="$ p < 8000$">.