一类不定方程及其素数解

A class of indefinite equations and their prime solutions

运用初等数论中的同余理论以及组合方法讨论了方程(x3-1λ)/(x-1λ)=(yn-2λ)/(y-2λ)适合x>y>1,n>3且2■的整数解(x,y,n,1λ,2λ),1λ、2λ∈{±1}.证明了:该方程仅有解(x,y,n,1λ,2λ)=(5,2,5,1,1)和(7,2,7,-1,-1)可使x是素数或者素数的方幂.

Let λ1,λ2∈{±1}.Using congruent theory in elementary number theory and combinational method,the integer solutions（x,y,n,λ1,λ2） of the equation（x3-λ1）/（x-λ1）=（yn-λ2）/（y-λ2） satisfying xy1,n3 and 2 n are discussed.It is proved that the equation has only the solutions（x,y,n,λ1,λ2）=（5,2,5,1,1） and（7,2,7,-1,-1） with x is a prime or a prime power.