关于丢番图方程x^y+y^x=z^z的一点注记

Note on the Diophantine Equation x^y+ y^x= z^z

设x>1,y>1,z>1是方程xy+yx=zz的正整数解.利用p-adic对数线性型估计,证明了该方程的解满足:(i)当2∣y时,z<1.46×105且z≡1(mod8);(ii)当2|z时,z<6.27×105;(iii)当2|x时,z<2.68×109.改进了文[5]的结论.

Let x > 1,y > 1,z > 1 be positive integer solutions of the equation xy+ yx= zz. Using the linear forms in p-adic logarithm,it is proved that:( i) if 2 ∣ y,then z < 1. 46 × 105 and z ≡1( mod8);( ii) if 2 ∣ z,then z < 6. 27 × 105;( iii) if 2 ∣ x,then z < 2. 68 × 109.