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Upper bounds for the prime divisors of Wendt's determinant


Author: Anastasios Simalarides
Journal: Math. Comp. 71 (2002), 415-427
MSC (2000): Primary 11C20; Secondary 11Y40, 11D79
DOI: https://doi.org/10.1090/S0025-5718-00-01292-8
Published electronically: October 18, 2000
MathSciNet review: 1863011
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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.


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Additional Information

Anastasios Simalarides
Affiliation: T.E.I. of Chalcis, Psahna 34400, Euboea, Greece

DOI: https://doi.org/10.1090/S0025-5718-00-01292-8
Keywords: Wendt's determinant, Fermat's congruence
Received by editor(s): April 13, 1999
Received by editor(s) in revised form: February 24, 2000
Published electronically: October 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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