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  • $\begingroup$ thanks for the solution but the timing in my larger 2K numbers are the same as Solve $\endgroup$ Commented Dec 23, 2013 at 19:29
  • $\begingroup$ My machine still runs Cornacchia faster than Solve on the larger numbers but, as I said, your timings may vary. The FactorInteger method of @DanielLichtblau should not be dismissed so quickly; it is significantly faster than all so far. Perhaps if you could optimize the Cornacchia algorithm, since its inner workings are visible and not a black box like Solve, then we would all benefit. $\endgroup$ Commented Dec 23, 2013 at 20:16
  • $\begingroup$ I've added new findings $\endgroup$ Commented Dec 23, 2013 at 21:44
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    $\begingroup$ Here is a shorter variant. With[{r = PowerMod[-1, 1/2, p]}, GCD[p, r + I]]. But it's not any faster as far as I can tell. $\endgroup$ Commented Dec 23, 2013 at 22:43
  • $\begingroup$ @DanielLichtblau Thanks! Your PowerMod formulation is compact and much, much faster than the awkward JacobiSymbol equivalent I was using in another non-Cornacchia approach. $\endgroup$ Commented Jan 19, 2014 at 1:13