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

Johnson, Wells

doi:10.1090/S0025-5718-1977-0447108-8

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

Author: Wells Johnson
Journal: Math. Comp. 31 (1977), 519-526
MSC: Primary 10B15; Secondary 10A10
DOI: https://doi.org/10.1090/S0025-5718-1977-0447108-8
MathSciNet review: 0447108
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Abstract: Solutions to the congruences ${(1 + a)^{{p^n}}} \equiv 1 + {a^{{p^n}}}\pmod {p^{n + 2}}$ and ${(1 + s)^p} \equiv 1 + {s^p}\pmod {p^n}$ 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 $p \equiv 1\;\pmod 6$. They are derived from the existence of a primitive cube root of unity $\pmod p$. Constructive techniques for finding numerical examples are presented. The results are obtained by examining the p-adic expansions of the p-adic $(p - 1)$st roots of unity.

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