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

Dilcher, Karl; Skula, Ladislav

doi:10.1090/S0025-5718-1995-1248969-6

A new criterion for the first case of Fermat’s last theorem
HTML articles powered by AMS MathViewer

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

Takashi Agoh, On the Kummer-Mirimanoff congruences, Acta Arith. 55 (1990), no. 2, 141–156. MR 1061635, DOI 10.4064/aa-55-2-141-156
Gert Almkvist and Arne Meurman, Values of Bernoulli polynomials and Hurwitz’s zeta function at rational points, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), no. 2-3, 104–108. MR 1112244
Krystyna Bartz and Jerzy Rutkowski, On the von Staudt-Clausen theorem, C. R. Math. Rep. Acad. Sci. Canada 15 (1993), no. 1, 46–48. MR 1214216
Paul T. Bateman, George B. Purdy, and Samuel S. Wagstaff Jr., Some numerical results on Fekete polynomials, Math. Comput. 29 (1975), 7–23. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR 0480293, DOI 10.1090/S0025-5718-1975-0480293-9
A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969) Academic Press, London, 1971, pp. 213–222. MR 0314736
Petr Cikánek, A special extension of Wieferich’s criterion, Math. Comp. 62 (1994), no. 206, 923–930. MR 1216257, DOI 10.1090/S0025-5718-1994-1216257-9
Don Coppersmith, Fermat’s last theorem (case $1$) and the Wieferich criterion, Math. Comp. 54 (1990), no. 190, 895–902. MR 1010598, DOI 10.1090/S0025-5718-1990-1010598-2

History of the theory of numbers

Eine neue Gattung zahlentheoretischer Funktionen welche von zwei Elementen abhängen und durch gewisse lineare Funktional-Gleichungen definiert werden

On the residues of

to modulus

etc.

32

Andrew Granville and Michael B. Monagan, The first case of Fermat’s last theorem is true for all prime exponents up to 714,591,416,091,389, Trans. Amer. Math. Soc. 306 (1988), no. 1, 329–359. MR 927694, DOI 10.1090/S0002-9947-1988-0927694-5
Andrew Granville, The Kummer-Wieferich-Skula approach to the first case of Fermat’s last theorem, Advances in number theory (Kingston, ON, 1991) Oxford Sci. Publ., Oxford Univ. Press, New York, 1993, pp. 479–497. MR 1368443

Derivation of criteria for the first case of Fermat’s last theorem and the combination of these criteria to produce a new lower bound for the exponent

Einige Sätze über die aus den Wurzeln der Gleichung

gebildeten complexen Zahlen für den Fall, dass die Klassenanzahl durch

theilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat’schen Lehrsatzes

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 10.2307/1968791
M. Lerch, Zur Theorie des Fermatschen Quotienten $\frac {{a^{p-1}-1}}{p}=q(a).$, Math. Ann. 60 (1905), no. 4, 471–490 (German). MR 1511321, DOI 10.1007/BF01561092

Sur le dernier théorème de Fermat

150

Peter L. Montgomery, New solutions of $a^{p-1}\equiv 1\pmod {p^2}$, Math. Comp. 61 (1993), no. 203, 361–363. MR 1182246, DOI 10.1090/S0025-5718-1993-1182246-5

Über den grossen Fermat’schen Satz

126

Paulo Ribenboim, 13 lectures on Fermat’s last theorem, Springer-Verlag, New York-Heidelberg, 1979. MR 551363, DOI 10.1007/978-1-4684-9342-9
Paulo Ribenboim, The book of prime number records, Springer-Verlag, New York, 1988. MR 931080, DOI 10.1007/978-1-4684-9938-4
Ladislav Skula, On the Kummer’s system of congruences, Comment. Math. Univ. St. Paul. 35 (1986), no. 2, 137–163. MR 864734
Ladislav Skula, Some consequences of the Kummer system of congruences, Comment. Math. Univ. St. Paul. 39 (1990), no. 1, 19–40. MR 1059524
Ladislav Skula, Fermat’s last theorem and the Fermat quotients, Comment. Math. Univ. St. Paul. 41 (1992), no. 1, 35–54. MR 1166223
Zhi Hong Sun and Zhi Wei Sun, Fibonacci numbers and Fermat’s last theorem, Acta Arith. 60 (1992), no. 4, 371–388. MR 1159353, DOI 10.4064/aa-60-4-371-388

Sur une propriété des nombres premiers qui se rattache au théorème de Fermat

52

Jonathan W. Tanner and Samuel S. Wagstaff Jr., New bound for the first case of Fermat’s last theorem, Math. Comp. 53 (1989), no. 188, 743–750. MR 982371, DOI 10.1090/S0025-5718-1989-0982371-4

Extension of the criteria of Wieferich and Mirimanoff in connection with Fermat’s last theorem

144

I. M. Vinogradov, Elements of number theory, Dover Publications, Inc., New York, 1954. Translated by S. Kravetz. MR 0062138
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2

Zum letzten Fermat’schen Theorem

136

Andrew Granville and Zhi-Wei Sun, Values of Bernoulli polynomials, Pacific J. Math. 172 (1996), no. 1, 117–137. MR 1379289, DOI 10.2140/pjm.1996.172.117