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    $\begingroup$ Could use FactorInteger[m, GaussianIntegers->True]. If m is a prime of the form 4n+1 then this will factor it as a product of conjugate Gaussian primes. $\endgroup$ Commented Dec 23, 2013 at 19:22
  • $\begingroup$ @DanielLichtblau, Timing is the same $\endgroup$ Commented Dec 23, 2013 at 19:28
  • $\begingroup$ Probably means Solve is doing essentially the same thing under the hood. $\endgroup$ Commented Dec 23, 2013 at 19:47