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On Wendt’s determinant and Sophie Germain’s theorem. (English) Zbl 0797.11035

For an integer \(n>1\), let \(E_ n\) denote the set of all primes \(p>2\) such that \(np+1\) is prime and the Fermat equation \(x^ p + y^ p = z^ p\) has a solution with \(p \nmid xyz\). By Sophie Germain’s theorem, \(E_ 2\) is empty. If \(p \in E_ n\), then it is known that either \(p \mid n\) or \(np+1\) divides the resultant \(R_ n\) of the polynomials \(X^ n-1\) and \((-1-X)^ n-1\), the so-called Wendt’s determinant. Starting with this result, the authors show that \(E_ n\) is empty for all \(n \not \equiv 0 \pmod 3\) up to 500. They in fact compute the prime factors of \(R_ n\) for these \(n\) and observe that among them it is enough to consider those primes \(np+1\) for which \(p>n\) and \(mp+1\) is composite whenever \(m<n\) and \(3 \nmid m\). For \(n \leq 500\) there are just two such primes; it is conjectured by the authors that their natural density, for \(n \to \infty\), is zero.

MSC:

11D41 Higher degree equations; Fermat’s equation

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