## 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

### Keywords:

Fermat equation; resultant; Wendt’s determinant

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### References:

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