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 < p – M.
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).
Read more about this topic: Primitive Root Modulo n
Famous quotes containing the words order, magnitude, primitive and/or roots:
“A. Well, an old order is a violent one.
This proves nothing. Just one more truth, one more
Element in the immense disorder of truths.”
—Wallace Stevens (1879–1955)
“He all their ammunition
And feats of war defeats
With plain heroic magnitude of mind
And celestial vigour armed;”
—John Milton (1608–1674)
“The inability to control our children’s behavior feels the same as not being able to control it in ourselves. And the fact is that primitive behavior in children does unleash primitive behavior in mothers. That’s what frightens mothers most. For young children, even when out of control, do not have the power to destroy their mothers, but mothers who are out of control feel that they may destroy their children.”
—Elaine Heffner (20th century)
“April is the cruellest month, breeding
Lilacs out of the dead land, mixing
Memory and desire, stirring
Dull roots with spring rain.”
—T.S. (Thomas Stearns)