Computation of the least primitive root

Abstract

Let g ( p ) g(p) denote the least primitive root modulo p p , and h ( p ) h(p) the least primitive root modulo p 2 p^2 . We computed g ( p ) g(p) and h ( p ) h(p) for all primes p ≤ 10 16 p\le 10^{16} . As a consequence we are able to prove that g ( p ) > p 5 / 8 g(p)>p^{5/8} for all primes p > 3 p>3 and that h ( p ) > p 2 / 3 h(p)>p^{2/3} for all primes p p . More generally, we provide values of p α p_\alpha where g ( p ) > p α g(p)>p^\alpha when p > p α p>p_\alpha , for various values of α \alpha with 1 / 2 > α > 5 / 8 1/2>\alpha >5/8 . Additionally, we give a log-histogram of g ( p ) g(p) when g ( p ) ≥ 100 g(p)\ge 100 and empirical evidence that g ( p ) ≪ ( log ⁡ p ) ( log ⁡ log ⁡ p ) 2 g(p)\ll (\log p)(\log \log p)^2 .