On Wendt's Determinant and Sophie Germain's Theorem

Research supported by the Natural Sciences and Engineering Research Council (Canada) and Fonds pour la Formation de Chercheurs et l'Aide a la Recherche (Quebec). Some results in section 3 of this work are taken from Jha's Ph.D. thesis [Jha 1992]. 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 resultantRn of the polynomialsXn 1 and ( 1 X)n 1, called Wendt’s determinant. The investigation of a conjecture about essential prime factors of Rn (Conjecture 1.3) leads to a proof that Case I of Fermat’s Last Theorem holds for any prime exponent p > 2 such that np + 1 is prime for some integer n 500 not divisible by 3.