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

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