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:
“It is the international system of currency which determines the totality of life on this planet. That is the natural order of things today. That is the atomic, and sub-atomic, and galactic
structure of things today. And you have meddled with the primal forces of nature! And you will atone! Am I getting through to you, Mr. Beale?”
—Paddy Chayefsky (1923–1981)
“If our condition were truly happy, we would not need diversion from thinking of it in order to make ourselves happy.”
—Blaise Pascal (1623–1662)
“Constancy has nothing virtuous in itself, independently of the pleasure it confers, and partakes of the temporizing spirit of vice in proportion as it endures tamely moral defects of magnitude in the object of its indiscreet choice.”
—Percy Bysshe Shelley (1792–1822)
“The sight of a Black nun strikes their sentimentality; and, as I am unalterably rooted in native ground, they consider me a work of primitive art, housed in a magical color; the incarnation of civilized, anti-heathenism, and the fruit of a triumphing idea.”
—Alice Walker (b. 1944)
“Look at this poet William Carlos Williams: he is primitive and native, and his roots are in raw forest and violent places; he is word-sick and place-crazy. He admires strength, but for what? Violence! This is the cult of the frontier mind.”
—Edward Dahlberg (1900–1977)