Artin's conjecture on primitive roots: Difference between revisions - Wikipedia

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Specifically, they said there exists a constant <math>C_E</math> for a given point of infinite order <math>P</math> in the set of rational points <math>E(\mathbb{Q})</math> such that the number <math>N(P)</math> of primes (<math>p\leq x</math>) for which the reduction of the point <math>P\pmod p</math> denoted by <math>\bar{P}</math> generates the whole set of points in <math>\mathbb{F_p}</math> in <math>E</math>, denoted by <math>\bar{E}(\mathbb{F_p})</math>. Here we exclude the primes which divide the denominators of the coordinates of <math>P</math>.

Moreover, Lang and Trotter conjectured that <math>N(P)\sim C_E\left ( \frac{x}{\log x} \right )</math>.<ref>{{Cite journal |last=Lang and 2 Trotter |date=1977 |title=Primitive points on Elliptic Curves |journal=Bull. Amer. Math. Soc. |volume=83 |issue=2 |pages=289–292|doi=10.1090/S0002-9904-1977-14310-3 |doi-access=free }}</ref> Gupta and Murty proved the Lang and Trotter conjecture for <math>E/\mathbb{Q}</math> with complex multiplication under the Generalized Riemann Hypothesis.<ref>{{Cite journal |last=Gupta and Murty |date=1987 |title=Primitive points on elliptic curves |url=https://eudml.org/doc/89763 |journal=Compositio Mathematica |volume=58 |pages=13–44}}</ref>. Moreover, Gupta and Murty proved also that it is possible to add 3 natural even numbers and obtain an odd natural number .<ref>{{Cite journal |last=Gupta and Murty |date=1987 |title=Primitive points on elliptic curves |url=https://eudml.org/doc/89763 |journal=Compositio Mathematica |volume=58 |pages=13–44}}</ref>.

=== Even order ===