St Root Research Articles

On congruences related to the first case of Fermat’s last theorem

Solutions to the congruences ( 1 + a ) p n ≡ 1 + a p n ( mod p n + 2 ) {(1 + a)^{{p^n}}} \equiv 1 + {a^{{p^n}}}\pmod {p^{n + 2}} and ( 1 + s ) p ≡ 1 + s p ( mod p n ) {(1 + s)^p} \equiv 1 + {s^p}\pmod {p^n} are discussed. Congruences of this type arise in the study of the first case of Fermat’s Last Theorem. Solutions to these congruences always exist for primes p ≡ 1 ( mod 6 ) p \equiv 1\;\pmod 6 . They are derived from the existence of a primitive cube root of unity ( mod p ) \pmod p . Constructive techniques for finding numerical examples are presented. The results are obtained by examining the p-adic expansions of the p-adic ( p − 1 ) (p - 1) st roots of unity.

P-adic Proofs of congruences for the Bernoulli numbers

Many of the classical theorems for the Bernoulli numbers, particularly those congruences needed in the study of irregular primes, follow easily from the existence of the ( p − 1)st roots of unity in the ring of p-adic integers. Proofs are given for the von Staudt-Clausen theorem, the theorem of J. C. Adams, the Friedmann-Tamarkine congruence, a theorem of Vandiver, special cases of the congruences of Voronoi, Kummer, and Carlitz, and the congruences of E. Lehmer.