不定方程x(x+1)(x+2)(x+3)=42y(y+1)(y+2)(y+3)的正整数解

On the positive integer solution of Diophantine equation x(x+1)(x+2)(x+3)=42y(y+1)(y+2)(y+3)

当M,N为给定正整数时,为解决不定方程Mx(x+1)(x+2)(x+3)=Ny(y+1)(y+2)(y+3)的求解问题,利用Pell方程基本解性质,同余思想以及递归数列等初等方法得到并证明了在(M,N)=(1,42)时该不定方程仅有正整数解(x,y)=(7,2)。进一步完善了当M=1时,N在50以内有正整数解的情形。

As M,N are both given positive integer,to solve the resolution problem about the Diophantine equation Mx(x+1)(x+2)(x+3)=Ny(y+1)(y+2)(y+3).Based on the basic solution of Pell equation, recursive sequence,congruence theory and other elementary methods,it is proved that when ( M,N )=(1,42) the Diophantine equation has only one positive integer solution,that is (x,y)=(7,2).Further improving that N ≤50 has positive integer solution when M =1.