Primitive Root Modulo n - Order of Magnitude of Primitive Roots

Order of Magnitude of Primitive Roots

The least primitive root modulo p is generally small.

Let gp be the smallest primitive root modulo p in the range 1, 2, ..., p–1.

Fridlander (1949) and Salié (1950) proved that there is a positive constant C such that for infinitely many primes gp > C log p.

It can be proved in an elementary manner that for any positive integer M there are infinitely many primes such that M < gp < pM.

Burgess (1962) proved that for every ε > 0 there is a C such that

Grosswald (1981) proved that if, then .

Shoup (1990, 1992) proved, assuming the generalized Riemann hypothesis, that gp =O(log6 p).

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