Diophantine方程组a^2+b^2=c^r和a^x+b^y=c^z的一点注记

A Note on the Diophantine System a^2+b^2=c^r and a^x+b^y=c^2

设r是大于1的正奇数,m是正偶数,V(r)+U(r)(-1)^(1/2)=(m+(-1)^(1/2))~r.本文证明了:当a=|V(r)|,b=|U(r)|,c=m^2+1时,如果r≡5(mod8),m>r^2且r<11500或者m>2r/π且r>11500,则方程a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,r).

Let r be a positive odd integer with r 〉 1, and let m be a positive even integer. Further move let a = ｜V（r）｜, b = ｜g（r）｜ and c = m^2 ＋ 1, where V（r） ＋ V（r）v√-1 = （m ＋ √-1）^r. In this paper we prove that if r ≡ 5 （mod8） and either m 〉 r^2, r 〈 11500 or m 〉 2r/π, r 〉 11500, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2, 2, r）.