Determine the minimum value of a positive integer constant K such that the equation:
X*Y^{2}  Y^{2} + 2*X + Y = K has precisely four solutions in positive integers.
In my precedent post I've found five solutions for K=60. When I posted I didn't notice that there is always a trivial solution for x=1 as then
y+2=K and there is always a pair of integers (x,y) = (1, k2) for each possible value of K. So I find yself with five solutions instead of four.
But as the problem ask for precisely four solutions I've been trying to find another more fitting answer. The process is quite time consuming (to do it properly) with the way I followed, so I have to give up without been sure about the result.
The lowest set of four solutions I have found are:
K=258; (x,y) = (86,1) (24,3) (15,4) (1,256)
(the last is the "trivial solution")
As you can see, K here is higher than K=60 (which has five solutions)

Posted by armando
on 20160427 06:00:14 