On Wendt's determinant and Sophie Germain's theorem

Abstract

After a brief review of partial results regarding CaseI 1 of Fermat's Last Theorem, we discuss the relationship between the number of points on Fermat's curve modulo a prime and the resultant $R_n$ of the polynomials $X^n - 1$ and $(-1-X)^n - 1$, called Wendt's determinant. The investigation of a conjecture about essential prime factors of $R_n$ (Conjecture 1.3) leads to a proof that Case 1 of Fermat's Last Theorem holds for any prime exponent $p>{}$2 such that $np+{}$1 is prime for some integer $n\le{}$500 not divisible by 3.

EDITOR'S NOTE: In addition to providing insight into Wendt's determinant, an object of interest in its own right, this paper belongs to a continuing line of investigations that may prove fruitful in spite of the recent announcement by Wiles of his proof of Fermat's Last Theorem. It is not unreasonable to hope for a more elementary proof than Wiles'.