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:
“Man is clearly made to think. It is his whole dignity and his whole merit; and his whole duty is to think as he ought. And the order of thought is to begin with ourselves, and with our Author and our end.”
—Blaise Pascal (1623–1662)
“It is not enough for theory to describe and analyse, it must itself be an event in the universe it describes. In order to do this theory must partake of and become the acceleration of this logic. It must tear itself from all referents and take pride only in the future. Theory must operate on time at the cost of a deliberate distortion of present reality.”
—Jean Baudrillard (b. 1929)
“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)
“Perhaps our own woods and fields,—in the best wooded towns, where we need not quarrel about the huckleberries,—with the primitive swamps scattered here and there in their midst, but not prevailing over them, are the perfection of parks and groves, gardens, arbors, paths, vistas, and landscapes. They are the natural consequence of what art and refinement we as a people have.... Or, I would rather say, such were our groves twenty years ago.”
—Henry David Thoreau (1817–1862)
“If church prelates, past or present, had even an inkling of physiology they’d realise that what they term this inner ugliness creates and nourishes the hearing ear, the seeing eye, the active mind, and energetic body of man and woman, in the same way that dirt and dung at the roots give the plant its delicate leaves and the full-blown rose.”
—Sean O’Casey (1884–1964)