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 of, order, magnitude, primitive and/or roots:
“In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
—Jorge Luis Borges (1899–1986)
“Judicial judgment must take deep account ... of the day before yesterday in order that yesterday may not paralyze today.”
—Felix Frankfurter (1882–1965)
“Government is either organized benevolence or organized madness; its peculiar magnitude permits no shading.”
—John Updike (b. 1932)
“The mountainous region of the State of Maine stretches from near the White Mountains, northeasterly one hundred and sixty miles, to the head of the Aroostook River, and is about sixty miles wide. The wild or unsettled portion is far more extensive. So that some hours only of travel in this direction will carry the curious to the verge of a primitive forest, more interesting, perhaps, on all accounts, than they would reach by going a thousand miles westward.”
—Henry David Thoreau (1817–1862)
“Though of erect nature, man is far above the plants. For man’s superior part, his head, is turned toward the superior part of the world, and his inferior part is turned toward the inferior world; and therefore he is perfectly disposed as to the general situation of his body. Plants have the superior part turned towards the lower world, since their roots correspond to the mouth, and their inferior parts towards the upper world.”
—Thomas Aquinas (c. 1225–1274)