Primitive Roots

As a consequence of [McGown and Trudgian, 2020], we find the next result.

Theorem (2020)

The least primitive root \(g(p)\) modulo the prime \(p\) satisfies \(g(p)\le p^{5/8}\) when \(p\ge 10^{22}\) and \(g(p) < \sqrt{p}-2\) when \(p\ge 10^{56}\).

From [McGown et al., 2016], we also have:

Theorem (2016)

Under GRH, the least primitive root \(g(p)\) modulo the prime \(p\) satisfies \(g(p) 409\).

Similar investigations concerning primivite roots modulo \(p^2\) are led in [Kerr et al., 2020] and in [Chen, 2022] where the next theorem is proved.

Theorem (2022)

The least primitive root \(h(p)\) modulo \(p^2\) satisfies \(h(p)\le p^{0.74}\) for all \(p \ge 2\).