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A New Criterion for the First Case of Fermat's Last Theorem

Karl Dilcher and Ladislav Skula
Mathematics of Computation
Vol. 64, No. 209 (Jan., 1995), pp. 363-392
DOI: 10.2307/2153341
Stable URL: http://www.jstor.org/stable/2153341
Page Count: 30
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A New Criterion for the First Case of Fermat's Last Theorem
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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 mod l for all integers N and k with 1 ≤ N ≤ 46 and 0 ≤ k ≤ N - 1. This is equivalent to $B_{l - 1}(k/N) - B_{l - 1} \equiv 0 (\operatorname{mod} l)$, where Bn and Bn(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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