Note on a Fermat-type diophantine equation

Abstract

Let p>5 be a prime number and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1, (k,p)=1 . Let C p ( i) be the eigenspace of the p-Sylow subgroup of ideal class group C of Q (ζ) corresponding to ω i, ω being the Teichmuller character. In this article we extend the main theorem in Sitaraman (J. Number Theory 80 (2000) 174) and get the following: For any fixed odd positive integer n< p−4, assume: (a) At least one of C p (3), C p (5),…, C p ( n) is non-trivial. (b) C p ( i) =0 for p− n−1⩽ i⩽ p−2. (c) 2 i≢1 mod (p) for 1⩽ i⩽ n+1. Let q be an odd prime such that q≡1 mod (p) , and such that there is a prime ideal Q over q in Q (ζ) whose ideal class is of the form I p J where J is non-trivial, not a pth power and J∈ C p (3)⊕ C p (5)⊕⋯⊕ C p ( n) . For such p and q, if x p + y p = pcz p has a non-trivial solution x,y,z∈ Z , with ( x, y, z)=1, then q∤ pcz p x+y . Let t( n)= n 224 n 4 . If log p>t(n) , then applying a result of Soulé (J. Reine Angew. Math. 517 (1999) 209), we show that the above result holds with only condition (a) because the others are automatically satisfied. We also make a remark about the effect of Soulé's result on the p-divisibility of h p + (the class number of the maximal real subgroup of Q (ζ) ) which is relevant to the existence of integral solutions to x p + y p = pcz p .