> **Possible Duplicate:**  
> [Primes P such that ((P-1)/2)!=1 mod P](http://mathoverflow.net/questions/16141/primes-p-such-that-p-1-21-mod-p)  


Motivation comes from comments in [this question](http://mathoverflow.net/questions/30101/composite-pairs-of-the-form-n-1-and-n1), and it is interesting in its own right. These primes are sequence [A055939](http://www.research.att.com/~njas/sequences/A055939) in OEIS.

So, which primes $p$ satisfy $p\\ |\\ (\frac{p-1}{2})! + 1$?

If my calculations (in sage) are correct, the following is true for all primes under 100,000. For $p > 3$:
$$p\\ |\\ (\frac{p-1}{2})! + 1 \iff h(\sqrt{-p})=1 \mod{4}$$