Three summation criteria for Fermat’s last theorem

Author:
H. Schwindt

Journal:
Math. Comp. **40** (1983), 715-716

MSC:
Primary 10-04; Secondary 10B15

DOI:
https://doi.org/10.1090/S0025-5718-1983-0689484-X

MathSciNet review:
689484

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Abstract: This paper extends the search for solutions of the congruences \[ \sum \limits _1^{[p/6]} {\frac {1}{i} \equiv 0,} \quad \sum \limits _1^{[p/6]} {\frac {1}{{{i^2}}} \equiv 0} \quad {\text {and}}\quad \sum \limits _{[p/6] + 1}^{[p/5]} {\frac {1}{i} \equiv 0\;\pmod p} \] to the limit $p < 600000$. The only solutions found were $p = 61$ in the first case, in the second $p = 205129$, and in the third case $p = 109$ and $p = 491$.

- Paulo Ribenboim,
*13 lectures on Fermat’s last theorem*, Springer-Verlag, New York-Heidelberg, 1979. MR**551363**
H. S. Vandiver "A new type of criteria for the first case of Fermat’s Last Theorem," - Emma Lehmer,
*On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson*, Ann. of Math. (2)**39**(1938), no. 2, 350–360. MR**1503412**, DOI https://doi.org/10.2307/1968791 - Donald E. Knuth,
*The art of computer programming. Vol. 2*, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR**633878** - D. H. Lehmer,
*On Fermat’s quotient, base two*, Math. Comp.**36**(1981), no. 153, 289–290. MR**595064**, DOI https://doi.org/10.1090/S0025-5718-1981-0595064-5

*Ann. of Math.*, v. 26, 1925, pp. 88-94. Schwindt, "Eine Bemerkung zu einem Kriterium von H. S. Vandiver,"

*Jahresbericht d. Deutschen Math. Verein*, v. 43, 1933-34, pp. 229-232.

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Article copyright:
© Copyright 1983
American Mathematical Society