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:
“Interpretation is the revenge of the intellect upon art. Even more. It is the revenge of the intellect upon the world. To interpret is to impoverish, to deplete the worldin order to set up a shadow world of meanings.”
—Susan Sontag (b. 1933)
“We quaff the cup of life with eager haste without draining it, instead of which it only overflows the brimobjects press around us, filling the mind with their magnitude and with the throng of desires that wait upon them, so that we have no room for the thoughts of death.”
—William Hazlitt (17781830)
“Human beings will be happiernot when they cure cancer or get to Mars or eliminate racial prejudice or flush Lake Erie but when they find ways to inhabit primitive communities again. Thats my utopia.”
—Kurt Vonnegut, Jr. (b. 1922)
“There is nothing but is related to us, nothing that does not interest us,kingdom, college, tree, horse, or iron show,the roots of all things are in man.”
—Ralph Waldo Emerson (18031882)