I'm stuck in a thought experiment: can some primitive root (or, for that matter, at least one)? $ w $ of $ N $ be expressed as $ w = a ^ 2-b $, where $ (b | N) = – 1 $?
We know $ (w | N) = – 1 $ Y $ (a ^ 2 | N) = 1 $ From basic data of the Legendre symbol.
I feel that the answer should be yes (heuristically), as $ (b | N) = $ 1, and one can choose any $ a $, or for that matter, any primitive root. $ w $, for which there are $ phi (N-1) $ of them, but currently I do not know how to approach the issue.
Thank you 🙂