195 research outputs found
A two variable Artin conjecture
Let a and b be non-zero rational numbers that are multiplicatively
independent. We study the natural density of the set of primes p for which the
subgroup of the multiplicative group of the finite field with p elements
generated by (a\mod p) contains (b\mod p). It is shown that, under assumption
of the generalized Riemann hypothesis (GRH), this density exists and equals a
positive rational multiple of the universal constant S=\prod_{p
prime}(1-p/(p^3-1)). An explicit value of the density is given under mild
conditions on a and b. This extends and corrects earlier work of P.J. Stephens
(1976). Our result, in combination with earlier work of the second author,
allows us to deduce that any second order linear recurrence with reducible
characteristic polynomial having integer elements, has a positive density of
prime divisors (under GRH)
Constructing elliptic curves of prime order
We present a very efficient algorithm to construct an elliptic curve E and a
finite field F such that the order of the point group E(F) is a given prime
number N. Heuristically, this algorithm only takes polynomial time Otilde((\log
N)^3), and it is so fast that it may profitably be used to tackle the related
problem of finding elliptic curves with point groups of prime order of
prescribed size. We also discuss the impact of the use of high level modular
functions to reduce the run time by large constant factors and show that recent
gonality bounds for modular curves imply limits on the time reduction that can
be obtained.Comment: 13 page
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