Determine the minimum value of a positive integer constant c such that the equation xy³ - y³+ x + y = c has precisely four distinct solutions in positive integers.

Yes, I also got c = 200 and the points (1,199), (4,4), (8,3), (100,1), but by doing a computer search of a rectangular grid which I found as follows.

The equation can be rearranged to x = (ył + c - y)/(ył + 1). Then comparing the numerator and denominator gives:
For x>=1, c - y >=1, so y<=c - 1.
For y>=1, x<=c/2.
So, it follows that any solutions lie in the domain 1<=x<=c/2, 1<=y<=c-1 and this can conveniently limit the search.

As Daniel spotted, the corner (1, c-1) is always a solution and the corner (c/2, 1) is a solution when c is even.

 

Posted by Harry on 2009-05-02 22:57:28