关于不定方程6x(x+1)(x+2)(x+3)=7y(y+1)(y+2)(y+3)

On the Diophantine Equation 6x(x+1)(x+2)(x+3)=7y(y+1)(y+2)(y+3)

主要运用pell方程、递推序列、同余式及(非)平方剩余等一些初等方法,证明了不定方程6x(x+1)(x+2)(x+3)=7y(y+1)(y+2)(y+3)仅有正整数解(x,y)=(25,24).沿用该文相同思路和方法得出关于不定方程mx(x+1)(x+2)(x+3)=ny(y+1)(y+2)(y+3)其中(m,n)=(6,11)和(m,n)=(5,11)时均无正整数解.

Such elementary methods as Pell equation,recursive sequence,congruence and quadratic（non-）residue are mainly used to prove that the diophantine equation 6 x（x＋1）（x＋2）（x＋3）=7 y（y＋1）（y＋2）（y＋3）has only the positive integer solution（x,y）=（25,24）.Based on similar ideas and methods,this paper proves that the diophantine equation mx（x＋1）（x＋2）（x＋3）=ny（y＋1）（y＋2）（y＋3）has no positive integer solution when（m,n）=（6,11）or（m,n）=（5,11）.