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大家知道,n(n+1)/2(=1+2_…+n)个队员可以排成一个每边有n个人的三角队形(我们称这种数为三角数),但在某些时候,他們也能排成一正方的队形。例如当n=8时,8·9/2=36个队員既能排成一海边有8人的三角队形,又能排成一每边有6人的正方队形。又如当n=49时,49·50/2=1225个队員既能排成一海边有49人的三角队形,也能排成一每边有35人的正方队形。容易驗証:当n=288,1681,9800,…时,都有此性质。現在我們要求出具有这种性质的一切n来。显然,上面的問题就是要去求出不定方程 n(n+1)/2=m~2 (1)的一切整数解的問題。在这篇短文中,我們将要証明:不定方程(1)具有无穷多个整数解,并且它們都能通过一定的程序求出。
As you know, n(n+1)/2(=1+2_...+n) players can be arranged in a triangular formation with n individuals on each side (we call this number as a triangle), but in some At the time, they can also form a square formation. For example, when n=8, 8·9/2=36 players can be arranged in a triangle formation with 8 people on a seaside and a square formation with 6 people on each side. For example, when n=49, 49.50/2=1,225 players can line up to form a triangle with 49 people on one side of the sea, and can also form a square formation with 35 people on each side. Easy to verify: When n=288, 1681, 9800,..., it has this property. Now we ask for everything n with this nature. Obviously, the above problem is to solve the problem of all integer solutions to the indefinite equation n(n+1)/2=m~2 (1). In this short essay, we will prove that the indefinite equation (1) has infinitely many integer solutions, and that they can all be determined by a certain procedure.