My Math Forum Integer Solutions to Z²-sZ+p=0

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 January 3rd, 2014, 05:42 AM #1 Member   Joined: Aug 2012 Posts: 40 Thanks: 3 Integer Solutions to Z²-sZ+p=0 Hey there, guys! Let the equation Z²-sZ+p=0 with integer coefficients. Then, the equation has integer solutions if, and only if, s²-4p 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: z²-3187z+2539212 = 0 It has two solutions: 1588, 1599 But it may be a little boring to calculate s²-4p 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 z²-2kz+k²=0 then z=k; But those are too specific and not very helpfull at all... Also, in the last case checking that p = k² and s = 2k doesn't help for large integers...

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