Subproducts of small 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...

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

Exploring Properties of Residue Classes

We report on an experiment in exploring properties of residue classes over the integers with the combined e ort of a multi-strategy proof planner and two computer algebra systems. An exploration module classies a given set and a given operation in terms of the algebraic structure they form. It then calls the proof planner to prove or refute simple properties of the operation. Moreover, we use d...

Residue Classes Having Tardy Totients

We show, in an effective way, that there exists a sequence of congruence classes ak (mod mk) such that the minimal solution n = nk of the congruence φ(n) ≡ ak (mod mk) exists and satisfies log nk/ logmk → ∞ as k → ∞. Here, φ(n) is the Euler function. This answers a question raised in [3]. We also show that every congruence class containing an even integer contains infinitely many values of the ...