关于方程S_x(n)=S_y(3)

ON THE EQUATION S_x(n)=S_y(3)

对于正整数m、n(n≥ 3) ,设Sm(n)是第m个n角数 .证明了 :当n >6且n - 2是平方数时 ,方程Sx(n) =Sy(3)无正整数解 (x ,y) ;当n >6 ,2 n且n - 2非平方数时 ,该方程有无穷多组正整数解 (x,y) .

For any positive integers m,n with n≥3 ,Let S m(n) denote the m -th n -gonal number,that (ⅰ) if n >6 and n -2 was a square was proven,then the equation S x(n)=S y(3) has no positive integer solution (x,y) ;(ⅱ)if n >6,2 n and n-2 was not a square,then the equation has infinitely many positive integer solutions (x,y) .