On Wendt's Determinant and Sophie Germain's Theorem

Abstract

After a brief review of partial results regarding Case I of Fermat's Last Theorem, we discuss the relationship between the number of points on Fermat's curve modulo a prime and the resultant Rn 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 CaseI of Fermat's LastTheorem holds for any prime exponent p > 2 such that np + 1 is prime for some integer n ≤ 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'.