Question

求方程组x的立方－y的立方－z的立方＝3xyz,x的平方＝2(y+z)的正整数解.

Answer 1

1.x^2 = 2(y+z) 则x是偶数,设x=2n,得

2n^2 = (y+z)

2.x^3 = y^3 + z^3 +3xyz得

8n^3 = (y+z)^3 - 3yz(y+z) + 3xyz

= 8n^6 - 6n^2*yz + 6n*yz

=> n!=1 时,yz = 4n^2(n^2+n+1)/3

或者 n=1,x=2,y=z=1