Upper bounds for the prime divisors of Wendt’s determinant

Simalarides, Anastasios

doi:10.1090/S0025-5718-00-01292-8

Upper bounds for the prime divisors of Wendt’s determinant
HTML articles powered by AMS MathViewer

by Anastasios Simalarides;

Math. Comp. 71 (2002), 415-427

DOI: https://doi.org/10.1090/S0025-5718-00-01292-8

Published electronically: October 18, 2000

PDF | Request permission

Abstract:

Let $c\geq 2$ be an even integer, $(3,c)=1$. The resultant $W_c$ of the polynomials $t^c-1$ and $(1+t)^c-1$ is known as Wendt’s determinant of order $c$. We prove that among the prime divisors $q$ of $W_c$ only those which divide $2^c-1$ or $L_{c/2}$ can be larger than $\theta ^{c/4}$, where $\theta =2.2487338$ and $L_n$ is the $n$th Lucas number, except when $c=20$ and $q=61$. Using this estimate we derive criteria for the nonsolvability of Fermat’s congruence.

References

David W. Boyd, The asymptotic behaviour of the binomial circulant determinant, J. Math. Anal. Appl. 86 (1982), no. 1, 30–38. MR 649851, DOI 10.1016/0022-247X(82)90250-5
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff Jr., Factorizations of $b^n \pm 1$, 2nd ed., Contemporary Mathematics, vol. 22, American Mathematical Society, Providence, RI, 1988. $b=2,3,5,6,7,10,11,12$ up to high powers. MR 996414, DOI 10.1090/conm/022
John Brillhart, Peter L. Montgomery, and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), no. 181, 251–260, S1–S15. MR 917832, DOI 10.1090/S0025-5718-1988-0917832-6
S. Chowla, Some conjectures in elementary number theory, Norske Vid. Selsk. Forh. (Trondheim) 35 (1962), 13. MR 139563
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Greg Fee and Andrew Granville, The prime factors of Wendt’s binomial circulant determinant, Math. Comp. 57 (1991), no. 196, 839–848. MR 1094948, DOI 10.1090/S0025-5718-1991-1094948-8
David Ford and Vijay Jha, On Wendt’s determinant and Sophie Germain’s theorem, Experiment. Math. 2 (1993), no. 2, 113–120. MR 1259425
I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 81
Charles Helou, On Wendt’s determinant, Math. Comp. 66 (1997), no. 219, 1341–1346. MR 1423075, DOI 10.1090/S0025-5718-97-00870-3
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 1, DOI 10.1090/gsm/058
E. Lehmer, On a resultant connected with Fermat’s Last Theorem, Bull. Amer. Math. Soc. 41 (1935), 864–867.
Paulo Ribenboim, 13 lectures on Fermat’s last theorem, Springer-Verlag, New York-Heidelberg, 1979. MR 551363
A. Simalarides, Applications of the theory of cyclotomic field to Fermat’s equation and congruence, Ph.D. Thesis, Athens University, Athens 1984.
Anastasios Simalarides, Sophie Germain’s principle and Lucas numbers, Math. Scand. 67 (1990), no. 2, 167–176. MR 1096451, DOI 10.7146/math.scand.a-12327
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
E. Wendt, Arithmetische Studien über den letzten Fermatschen Satz, welcher aussagt, dass die Gleichung $a^n=b^n+c^n$, für $n>2$ in ganzen Zahlen nicht auflösbar ist, J. Reine Angew. Math. 113 (1894), 335–347.