Note on a Fermat-type diophantine equation |
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Authors: | Sankar Sitaraman |
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Institution: | Department of Mathematics, Howard University, Washington, DC 20059, USA |
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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 Cp(i) be the eigenspace of the p-Sylow subgroup of ideal class group C of corresponding to ωi,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 Cp(3),Cp(5),…,Cp(n) is non-trivial.
- (b)
- Cp(i)=0 for p−n−1?i?p−2.
- (c)
- for 1?i?n+1.
Let q be an odd prime such that , and such that there is a prime ideal Q over q in whose ideal class is of the form IpJ where J is non-trivial, not a pth power and J∈Cp(3)⊕Cp(5)⊕?⊕Cp(n).For such p and q, if xp+yp=pczp has a non-trivial solution , with (x,y,z)=1, then .Let t(n)=n224n4. If , 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 hp+ (the class number of the maximal real subgroup of ) which is relevant to the existence of integral solutions to xp+yp=pczp. |
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