关于方程S_x(n)=S_y(3)的商榷

On the Diophantus Equation S_x(n)=S_y(3)

与第m个n角数Sm(n)相联系的方程Sx(n) =Sy( 3) ,证明了 :( 1 )当D =n -2是非平方数 ,且u12 -Dv12 =-1有解 (u1,v1)时 ,则该方程有无穷多组解 .( 2 )当n-2是非平方数时 ,该方程或者无解或者有无穷多解 .举例说明了结论 ( 1 )中u12 -Dv12 =-1有解的条件不是必要的 .还指出了文献

The Diophantus equation S x(n)=S y(3) is relative to the m _th n _gonal number S m(n). This article proves that(1)If D=n-2 is not a square and equation u 1 2-Dv 1 2=-1 has a solution (u 1,v 1) ,then the first equation has infinitely many positive integer solutions (x,y) .(2)If n-2 is not a square,then the first equation has no solution or has infinitely many positive integer solutions (x,y) .We illustrate by example the condition u 1 2-Dv 1 2=-1 has a solution (u 1,v 1) is not necessary for the theorem(1),and point out some mistakes in [3].