关于不定方程5x(x 1)(x 2)(x 3) = 42y(y 1)(y 2)(y 3)

On the Diophantine Equation 5x(x 1)(x 2)(x 3) = 42y(y 1)(y 2)(y 3)

本文运用同余式、递推序列和Pell方程等初等方法,证明了不定方程5x(x 1)(x 2)(x 3) = 42y(y 1)(y 2)(y 3)仅有唯一正整数解(x, y) = (6, 3),并找出了该方程的所有整数解。

This article uses elementary methods such as congruence formula, recursive sequences, and Pell equation to prove that indefinite Diophantine equation 5x(x 1)(x 2)(x 3) = 42x(x 1)(x 2)(x 3) has a unique positive integer (x, y) = (6, 3). Also, All 20 groups of integer solutions of the equation are found.