关于方程S_x(n)=S_y(3)

On Equation S_x(n)=S_y(3)

对于整数m、n(n≥ 3) ,设Sm(n)是第m个n角数 .本文证明了 :当n >6且n - 2是平方数时 ,方程Sx(n) =Sy( 3)无正整数解 ;当n >6,2 +n且n - 2非平方数时 。

For any positive integers m, n with n≥3, let Sm(n) denote the m th n gonal number. In this paper the author proves that (i) if n>6 and n 2 is a square, then the equation Sx(n)=Sy(3) has no positive integer solution; (ii) if n>6, 2+n and n 2 are not square, then the equation has infinitely many positive integer solutions.