联立丢番图方程组a^2+b^2=c^3和a^x+b^y=c^z

On the Diophantine Equation System a^2+b^2=c^3 and a^x+b^y=c^z

设a,b,c,是给定的互素正整数。本文证明了如下结果:如果a^2+b^2=c^3,b是奇素数的幂,则丢番图方程a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,3)。

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime power, then the diophantine equation a^x＋b^y=c^z has unique positive integer solution （x, y, z） = （2, 2, 3）.