Products in Residue Classes

Products in Residue Classes

Abstract. We consider a problem of P. Erdős, A. M. Odlyzko and A. Sárkőzy about the representation of residue classes modulom by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results “on average” over moduli m and residue classes modulo m and somewhat strong...

Divisors in residue classes, constructively

Let r, s, n be integers satisfying 0 ≤ r < s < n, s ≥ n, α > 1/4, and gcd(r, s) = 1. Lenstra showed that the number of integer divisors of n equivalent to r (mod s) is upper bounded by O((α − 1/4)). We re-examine this problem; showing how to explicitly construct all such divisors and incidentally improve this bound to O((α−1/4)−3/2).

Counting primes in residue classes

We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x) can be used for computing efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1 less than x for several values of x up to 1020 and found a new region where π(x, 4, 3) is less than π(x, 4, 1) near x = 1018.

Classifying Isomorphic Residue Classes

We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using di erent proving techniques, which are implemented as strategies in a multi-strategy proof planner. We show how these techniques help to successfully derive proofs in our domain and ex...

Classifying Isomorphi Residue Classes