A new criterion for the first case of Fermat’s last theorem
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- by Karl Dilcher and Ladislav Skula PDF
- Math. Comp. 64 (1995), 363-392 Request permission
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.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp. 64 (1995), 363-392
- MSC: Primary 11D41; Secondary 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-1995-1248969-6
- MathSciNet review: 1248969