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:
“My ideas are a curse.
They spring from a radical discontent
with the awful order of things.
I play clown. I play carpenter. I play nurse.
I play witch.”
—Anne Sexton (1928–1974)
““Learn what is true, in order to do what is right,” is the summing up of the whole duty of man, for all who are unable to satisfy their mental hunger with the east wind of authority.”
—Thomas Henry Huxley (1825–95)
“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)
“Children can’t make their own rules and no child is happy without them. The great need of the young is for authority that protects them against the consequences of their own primitive passions and their lack of experience, that provides with guides for everyday behavior and that builds some solid ground they can stand on for the future.”
—Leontine Young (20th century)
“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)