关于丢番图方程X^2+4P^m-1=4P^n

On the Diophantine Equation x^2+4p^m-1=4p^n

本文用代数数论的方法,并结合计算机的计算,完全解决了一类不定方程x(?)+4p^m-1=4p^n p^m(?)2((?))主要结果有:不定方程(*)除了当 p^m=3和 p^m=5时分别有非平凡解(p^m,x,n)=(3,31,5) ,(5,599,7) 之外,仅有平凡解(x,n)=(1,m)和(2p^m-1,2m).

In this paper,by using the method of algebraic number Theory and the aid of a computer,we solve completely the Diophantine Equation x^2+4p^m-1=4p^n p^m(?)2 (*) The main result is the following:The Diophantine Equation(*)have only trivial solutions (x,n)=(1,m),(2p^m-1,2m),apart from p^m=3and p^m=5. In these exceptional cases,two non-trivial solutions(p^m,x^mn)=(3,31,5) , (5,599,7) are obtained.