Irregular Primes and Cyclotomic Invariants

The table of irregular primes less than 30000 has been computed and deposited in the UMT file. The fraction of irregular primes in this range is 0.3924, close to the heuristic prediction of $1 - {e^{ - 1/2}}$. Fermat’s Last Theorem has been verified for all prime exponents $p < 30000$, and the cyclotomic invariants ${\mu _p},{\lambda _p}$, and ${\nu _p}$ of Iwasawa have been completely determined for these primes. The computations show that for p in this range, ${\mu _p} = 0$ and the invariants ${\lambda _p}$ and ${\nu _p}$ both equal the index of irregularity of p.