# number theory: at least one primitive root \$ w \$ of \$ N \$ can be expressed as \$ a ^ 2-b \$, where \$ (b | N) = – 1 \$

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 🙂