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:
“An example is often a deceptive mirror, and the order of destiny, so troubling to our thoughts, is not always found written in things past.”
—Pierre Corneille (1606–1684)
“Sometimes you’re overwhelmed when a thing comes, and you do not realize the magnitude of the affair at that moment. When you get away from it, you wonder, did it really happen to you.”
—Marian Anderson (1902–1993)
“Not only do our wives need support, but our children need our deep involvement in their lives. If this period [the early years] of primitive needs and primitive caretaking passes without us, it is lost forever. We can be involved in other ways, but never again on this profoundly intimate level.”
—Augustus Y. Napier (20th century)
“Sensuality often accelerates the growth of love so much that its roots remain weak and are easily pulled up.”
—Friedrich Nietzsche (1844–1900)