If (mod 8), then n is a quadratic residue modulo for any
I don'T see how to prove this, maybe by induction on r ?
This was a fun problem. There may be a simpler answer, but this is what I came up with (eventually). And yeah, it's induction on r.
Suppose n is a quadratic residue mod , i.e. there exist integers a,k such that . If k is even, then k=2c and then and we're done, so assume k odd.
Then
Since k+a is even and if , we have that mod .
That just leaves the base case, which is trivial: is a quadratic residue mod 8.