Cyclicity and generation of points mod p on elliptic curves.
Distributional properties of powers of matrices
We apply the larger sieve to bound the number of matrices not having large order when reduced modulo the primes in an interval. Our motivation is the relation with linear recursive congruential generators. Basically our results establish that the probability of finding a matrix with large order modulo many primes drops drastically when a certain threshold involving the number of primes and the order is exceeded. We also study, for a given prime and a matrix, the existence of nearby non-similar...
Écarts entre nombres premiers successifs
Le théorème des nombres premiers dit que la distance entre deux nombres premiers consécutifs est, en moyenne, de l’ordre de . Récemment, D. Goldston, J. Pintz et C. Yıldırım sont parvenus à démontrer que la distance normalisée pouvait devenir arbitrairement petite, améliorant spectaculairement les résultats connus auparavant. Sous des hypothèses considérées comme raisonnables, ils parviennent à montrer que infiniment souvent. Leur méthode est une très jolie application d’idées inspirée par...
Eine neue Darstellung und neue Anwendungen der Viggo Brunschen Methode.
Erratum: "On the number of prime divisors of the order of elliptic curves modulo p" (Acta Arith. 117 (2005), 341-352)
E-symmetric numbers
A positive integer n is called E-symmetric if there exists a positive integer m such that |m-n| = (ϕ(m),ϕ(n)), and n is called E-asymmetric if it is not E-symmetric. We show that there are infinitely many E-symmetric and E-asymmetric primes.
Extreme values of symmetric power L-functions at 1
Four prime squares and powers of 2
Four squares of primes and 165 powers of 2
Four squares of primes and powers of 2
By developing the method of Wooley on the quadratic Waring-Goldbach problem, we prove that all sufficiently large even integers can be expressed as a sum of four squares of primes and 46 powers of 2.
Gaussian primes
Gauss's three squares theorem with almost prime variables
Imaginary quadratic fields with small odd class number
Improving on the Brun-Titchmarsh theorem
Integers with no large prime factors
Kloosterman sums with prime variable
Lagrange's Four Squares Theorem with almost prime variables.
Landau’s problems on primes
At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :(1)Are there infinitely many primes of the form ?(2)The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.(3)The Twin Prime Conjecture.(4)Does there exist always at least one prime between neighbouring squares?All these...
Lattice points in a circle: An improved mean-square asymptotics
Le crible à vecteurs