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Mathematics of Computation

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A new criterion for the first case of Fermat's last theorem

Authors: Karl Dilcher and Ladislav Skula
Journal: Math. Comp. 64 (1995), 363-392
MSC: Primary 11D41; Secondary 11Y50
MathSciNet review: 1248969
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Abstract: It is shown that if the first case of Fermat's last theorem fails for an odd prime l, then the sums of reciprocals modulo l, $ s(k,N) = \sum 1/j\;(kl/N < j < (k + 1)l/N)$ are congruent to zero $ \bmod\;l$ for all integers N and k with $ 1 \leq N \leq 46$ and $ 0 \leq k \leq N - 1$. This is equivalent to $ {B_{l - 1}}(k/N) - {B_{l - 1}} \equiv 0 \pmod l$, where $ {B_n}$ and $ {B_n}(x)$ are the nth Bernoulli number and polynomial, respectively. The work can be considered as a result on Kummer's system of congruences.

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Article copyright: © Copyright 1995 American Mathematical Society

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