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January 3rd, 2014, 06:42 AM   #1
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Integer Solutions to Z-sZ+p=0

Hey there, guys!
Let the equation Z-sZ+p=0 with integer coefficients. Then, the equation has integer solutions if, and only if, s-4p is a perfect square.
But is there an "easier" or "shorter" way to determine when the equation has or hasn't solutions?
For example, consider the equation:

z-3187z+2539212 = 0
It has two solutions: 1588, 1599

But it may be a little boring to calculate s-4p and later checking if it's a perfect square if both numbers are too high. So, is there an easier way?

So far, I only could derive the trivial:
if both s and p are odd, then there's no solution;
if p is prime, then there's a solution iff s = p+1;
if the equation is on the form z-2kz+k=0 then z=k;

But those are too specific and not very helpfull at all... Also, in the last case checking that p = k and s = 2k doesn't help for large integers...
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