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:
“In Russia there is an emigration of intelligence: émigrés cross the frontier in order to read and to write good books. But in doing so they contribute to making their fatherland, abandoned by spirit, into the gaping jaws of Asia that would like to swallow our little Europe.”
—Friedrich Nietzsche (1844–1900)
“My time has come.
There are twenty people in my belly,
there is a magnitude of wings,
there are forty eyes shooting like arrows,
and they will all be born.
All be born in the yellow wind.”
—Anne Sexton (1928–1974)
“Cannibalism to a certain moderate extent is practised among several of the primitive tribes in the Pacific, but it is upon the bodies of slain enemies alone; and horrible and fearful as the custom is, immeasurably as it is to be abhorred and condemned, still I assert that those who indulge in it are in other respects humane and virtuous.”
—Herman Melville (1819–1891)
“The greatest gifts you can give your children are the roots of responsibility and the wings of independence.”
—Denis Waitly. Quoted in The Winning Family, ch. 25, by Louise Hart (1987)