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:
“The confusion of emotions with behavior causes no end of unnecessary trouble to both adults and children. Behavior can be commanded; emotions can’t. An adult can put controls on a child’s behavior—at least part of the time—but how do you put controls on what a child feels? An adult can impose controls on his own behavior—if he’s grown up—but how does he order what he feels?”
—Leontine Young (20th century)
“We quaff the cup of life with eager haste without draining it, instead of which it only overflows the brim—objects 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 (1778–1830)
“The price we pay for the complexity of life is too high. When you think of all the effort you have to put in—telephonic, technological and relational—to alter even the slightest bit of behaviour in this strange world we call social life, you are left pining for the straightforwardness of primitive peoples and their physical work.”
—Jean Baudrillard (b. 1929)
“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)