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:
“Out of the slimy mud of words, out of the sleet and hail of verbal imprecisions,
Approximate thoughts and feelings, words that have taken the place of thoughts and feelings,
There springs the perfect order of speech, and the beauty of incantation.”
—T.S. (Thomas Stearns)
“Woman ... cannot be content with health and agility: she must make exorbitant efforts to appear something that never could exist without a diligent perversion of nature. Is it too much to ask that women be spared the daily struggle for superhuman beauty in order to offer it to the caresses of a subhumanly ugly mate?”
—Germaine Greer (b. 1939)
“He all their ammunition
And feats of war defeats
With plain heroic magnitude of mind
And celestial vigour armed;”
—John Milton (1608–1674)
“[How] the young . . . can grow from the primitive to the civilized, from emotional anarchy to the disciplined freedom of maturity without losing the joy of spontaneity and the peace of self-honesty is a problem of education that no school and no culture have ever solved.”
—Leontine Young (20th century)
“A poet must be a psychologist, but a secret one: he should know and feel the roots of phenomena but present only the phenomena themselves—in full bloom or as they fade away.”
—Ivan Sergeevich Turgenev (1818–1883)