On a Fermat-Type Diophantine Equation

✍ Scribed by Sankar Sitaraman

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 173 KB
• Volume: 80
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2438

✦ Synopsis

Let p>3 be an odd prime and 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() corresponding to | i , | being the Teichmuller character. Let B 2i denote the 2i th 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 x p + y p = pc z p . In particular, we prove the following: Assume p is irregular, and p | B p&3 . Let q be an odd prime such that q#1 (mod p), and there is a prime ideal Q over q in Q(`) whose ideal class generates C (3) p , which is known to be cyclic. If x p + y p = pc z p has nontrivial integer solutions, then we show that q |% ( pc z p Â(x+ y)). We also give proof of the unsolvability of the above equation for regular primes ( p>3), using the results of