We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite. 1

Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem

We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular group, there is a density one sequence of integers N for which its order (or period) modulo N is somewhat larger than √ N.

Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coecients, then the order of A modulo p is greater than p for all p in a density one subset of the primes.

Let E be an elliptic curve defined over Q. For any prime p of good reduction, let Ep be the reduction of E mod p. Denote by Np the cardinality of Ep(Fp), where Fp is the finite field of p elements. Let P (Np) be the greatest prime divisor of Np. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, P (Np)> p ϑ(p), where ϑ(p) is any function of p such that ϑ(p) → 0 as p → ∞. Moreover we show that for such E there is a positive proportion of primes p ≤ x for which P (Np)> p ϑ, where ϑ is any number less than ϑ0 = 1 − 1 2 prove the following. Let Γ be a free subgroup of rank r ≥ 2 of the group of rational points E(Q), and Γp be the reduction of Γ mod p, then for a positive proportion of primes p ≤ x, we have where ɛ> 0. e− 1 4 = 0.6105 · · ·. As an application of this result we |Γp |> p ϑ0−ɛ Keywords: Reduction mod p of elliptic curves, Elliptic curves over finite fields, Brun-Titchmarsh inequality in number fields, Bombieri-Vinogradov theorem in number fields, Abelian extensions of imaginary quadratic number fields. 2000 Mathematics Subject Classification. Primary 11G20, Secondary 11N37. 1

Abstract. Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N 1−ǫ for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p 1−ǫ for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N 1−ǫ for all N in a density one subset of the integers. 1.