On a Fermat-Type Diophantine Equation

Abstract

Let p>3 be an odd prime 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(i)p be the eigenspace of the ideal class group of Q(ζ) corresponding to ωi, ω being the Teichmuller character. Let B2i denote the 2ith Bernoulli number. In this article we apply the methods (following H. S. Vandiver (1934, Bull. Amer. Math. Soc., 118–123)) which were used by the author (1994, Ph.D. Thesis, California Institute of Technology) to prove a special case Fermat's Last theorem, to study the equation xp+yp=pczp. In particular, we prove the following: Assume p is irregular, and p∣Bp−3. Let q be an odd prime such that q≡1 (modp), and there is a prime ideal Q over q in Q(ζ) whose ideal class generates C(3)p, which is known to be cyclic. If xp+yp=pczp has nontrivial integer solutions, then we show that q∤(pczp/(x+y)). We also give proof of the unsolvability of the above equation for regular primes (p>3), using the results of H. S. Vandiver (1936, Monatsh. Math. Phys.43, 317–320).