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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Keywords:
Congruences,
Fermat’s Last Theorem,
<I>p</I>-adic roots of unity
Article copyright:
© Copyright 1977
American Mathematical Society