rationals and squares

[DrMike]

I'm looking for a characterisation of those rational numbers e and c for which 4e^2-(2c+4)e+c^2+1 is a square. I always find this kind of problem very frustrating. Any tips? Thanks!

 

[Deleted member 4993]

I'm looking for a characterisation of those rational numbers e and c for which 4e^2-(2c+4)e+c^2+1 is a square. I always find this kind of problem very frustrating. Any tips? Thanks!

I may be misinterpreting your problem, but the solution to me seems straight forward (from completing square or quadratic equation):
The condition for single solution (repeated) of a quadratic equation is:

(2c+4)[sup:23fpdgd8]2[/sup:23fpdgd8] = 4 * 4 * (c[sup:23fpdgd8]2[/sup:23fpdgd8] +1)

3c[sup:23fpdgd8]2[/sup:23fpdgd8] - 4c = 0

so the solution is

any rational number for 'e' and c = 0 or c = 4/3

Your choice of 'e' as a rational variable puzzles me though!!

 

[DrMike]

I don't require there to be a single e for the given c.