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:
“Perchance not he but Nature ailed,
The world and not the infant failed.
It was not ripe yet to sustain
A genius of so fine a strain,
Who gazed upon the sun and moon
As if he came unto his own,
And, pregnant with his grander thought,
Brought the old order into doubt.”
—Ralph Waldo Emerson (1803–1882)
“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)
“Each structure and institution here was so primitive that you could at once refer it to its source; but our buildings commonly suggest neither their origin nor their purpose.”
—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)