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)
““I have said no
To everything, in order to get at myself.
I have wiped away moonlight like mud....””
—Wallace Stevens (1879–1955)
“War is pillage versus resistance and if illusions of magnitude could be transmuted into ideals of magnanimity, peace might be realized.”
—Marianne Moore (1887–1972)
“It was a very primitive kind of harbor, where boats were drawn up amid the stumps,—such a one, methought, as the Argo might have been launched in.”
—Henry David Thoreau (1817–1862)
“Our roots are in the dark; the earth is our country. Why did we look up for blessing—instead of around, and down? What hope we have lies there. Not in the sky full of orbiting spy-eyes and weaponry, but in the earth we have looked down upon. Not from above, but from below. Not in the light that blinds, but in the dark that nourishes, where human beings grow human souls.”
—Ursula K. Le Guin (b. 1929)