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:
“I do not know if you remember the tale of the girl who saves the ship under mutiny by sitting on the powder barrel with her lighted torch ... and all the time knowing that it is empty? This has seemed to me a charming image of the women of my time. There they were, keeping the world in order ... by sitting on the mystery of life, and knowing themselves that there was no mystery.”
—Isak Dinesen [Karen Blixen] (1885–1962)
“Sometimes you’re overwhelmed when a thing comes, and you do not realize the magnitude of the affair at that moment. When you get away from it, you wonder, did it really happen to you.”
—Marian Anderson (1902–1993)
“Financiers are great mythomaniacs, their explanations and superstitions are those of primitive men; the world is a jungle to them. They perceive acutely that they are at the dawn of economic history.”
—Christina Stead (1902–1983)
““Jim,” she said earnestly, “if I was put down there in the middle of the night, I could find my way all over that little town; and along the river to the next town, where my grandmother lived. My feet remember all the little paths through the woods, and where the big roots stick out to trip you. I ain’t never forgot my own country.””
—Willa Cather (1873–1947)