sqrt(2) modulo a prime


We know that 2 is a square modulo any prime of the form 8n-1 and 8n+1.
In the case 8n-1, the sqrt's of 2 are explicitly 4^n and -4^n. Likewise
we know that -1 is a square mod p=4n+1 and there is the explicit
formula +-(2n)! for its square roots. Last, there is a tougher formula
(due to Jacobi) for the roots of xx+x+1 = 0 mod p when p is 1 mod 6.
But what about sqrt(2) mod p=8n+1? Has any explicit formula been found?
I suspected it was possible and that it would be easy enough to smoke
out, but I haven't succeeded. TIA.

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