Upper bounds for the prime divisors of Wendt's determinant (Q2759110)

Let \(c>0\) be an even integer. Wendt's determinant \(W_c\) is the resultant of the polynomials \(t^c-1\) and \((1+t)^c-1\). Assume that 3 does not divide \(c\), since otherwise \(W_c=0\). Among the factors of \(W_c\) there is always \(2^c-1\) and, for \(c \equiv 2 \pmod 4\), the Lucas number \(L_{c/2}\); these are called the principal factors. The author proves that, excluding the case \(c=20\), if \(q\) is a prime factor of \(W_c\) exceeding the bound \(\theta^{c/4}\), where \(\theta = 2.2487338\), then \(q\) divides a principal factor. This result is sharper than the author's previous result [Math. Scand. 67, 167-176 (1990; Zbl 0688.10015)]. By using existing factorization tables, the author has found all primes \(q > \theta^{c/4}\) dividing the principal factors of \(W_c\) for \(c\leq 662\). A table of them is given. NEWLINENEWLINENEWLINEConsider the Fermat congruence (*) \(x^p+y^p+z^p\equiv 0 \pmod q\), where \(p\) and \(q\) are odd primes. If either \(q\not\equiv 1 \pmod p\) or \(q\equiv 1 \pmod{3p}\), then this congruence has a nontrivial solution. In the remaining case it is known that such a solution exists if and only if \(q\mid W_{(q-1)/p}\). That enables the author to apply the result above to provide certain sufficient conditions for the existence of a nontrivial solution of (*). This improves upon several previous results, including a result of the author himself [op. cit.].