1,721,014 research outputs found
On the exponents of the group of points of an Elliptic curve over a finite field
No full textWe present a lower bound for the exponent of the group of rational points of an elliptic curve over a finite field. Earlier results considered finite fields where either is fixed or and is prime. Here, we let both and vary; our estimate is explicit and does not depend on the elliptic curve
SCHOLAR—a Scientific Celebration Highlighting Open Lines of Arithmetic Research Conference in Honour of M. Ram Murty’s Mathematical Legacy on his 60th Birthday October 15–17, 2013 Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada
No full textOn Hooley's Theorem with weights. Rend. Sem. Mat. Polit. Univ. Torino 53 No. 4 (1995) 375-388. *
No full textWe adapt Hooley’s proof that the Generalized Riemann Hypothesis implies
the Artin Conjecture for primitive roots to various other problems. We consider the sum P
px f(ip) where ip is the index of 2 modulo p and f is a given function. In various
cases we establish asymptotic formulas for such a sum and analyse the constants. While
we claim no originality, we outline the method to approach this problem in a fairly
general case
The r-rank Artin Conjecture.
No full textWe assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which can be generated by given multiplicatively independent numbers. In the case when the given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown-Zassenhaus (J. Number Theory 3 (1971), 306-309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to
On minimal sets of generators for primitive roots
No full textAbstractA conjecture of Brown and Zassenhaus (see [2]) states that the first log/? primes generate a primitive root (mod p) for almost all primes p. As a consequence of a Theorem of Burgess and Elliott (see [3]) it is easy to see that the first log2p log log4+∊p primes generate a primitive root (mod p) for almost all primes p. We improve this showing that the first log2p/ log log p primes generate a primitive root (mod p) for almost all primes p.</jats:p
An analogue of Artin’s conjecture for multiplicative subgroups of the rationals
No full textGiven Γ⊂Q∗ a multiplicative subgroup and m∈N+ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which ind p Γ = m, where ind p Γ = (p − 1)/|Γ p | and Γ p is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi–Pappalardi, Lenstra, Moree, Murata, Wagstaff, and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations
Review of the book "Elliptic Curves in Cryptography" by I. Blake, G. Seroussi, N. Smart.
No full text
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