商高数的Jesmanowicz猜想

On Jesmanowicz's Conjecture Concerning Pythagorean Numbers

利用Bilu、Hanrot和Voutier关于本原素因子存在性理论及二次丢番图方程解的表示方面的一些精细结果证明:当a=n+1,b=2n(n+1),c=2n(n+1)+1时,方程a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,2).

We show that when a = 2n＋1,b = 2n（n＋l）,c = 2n（n＋1）＋1 with n positive integer, the equation a^x ＋ b^y =c^z has only the positive integer solution （x, y, z） = （2, 2, 2）. The proof is based on using the theorem about the existence of primitive divisors of Lucas numbers due to Bilu, Hanrot ＆ Voutier and some fine results on the representation of the solutions of quadratic diophantine equations.