On Tuesday the 6th of September at 10 CET I will give (online) a talk at the ACAI conference 2022 (International Conference on Advances in Computing Technologies and AI), which takes place in Dubai from the 6th to the 8th of September.
The title of the talk is “On primality conditions and residue number systems“; an abstract follows.
I will present a research programme aimed at investigating generalized primality conditions through residue number systems. Sieve methods from analytic number theory have proved to be very effective tools in addressing problems concerning prime numbers, such as, most notably, the Goldbach’s conjecture. Still, these methods are mostly based on analytic estimations rather than on structural considerations about residue number systems. We propose to experimentally study, through suitable computer programs, complementary sets of solutions to systems of congruences, in order to eventually formulate theoretical conjectures about their behavior and obtain insights, in particular, on the difficult problem of effectively characterizing the natural order relation on numbers in terms of modular representations. This should lead to a unified framework in which different problems such as the Goldbach’s conjecture and the twin prime conjecture can be constructively investigated under a common roof as part of an abstract theory of generalized primality conditions.
The subject of residue number systems has fascinated me since my teenage years, when I started pondering about these issues and developing a unifying framework for investigating generalized primality conditions. This subject is actually strictly related to topos theory, as the central result in the theory, namely the Chinese Remainder Theorem, can be interpreted as some kind of sheaf condition.
I am convinced that this subject would greatly benefit from extensive experimentations on a computer aimed at formulating theoretical conjectures about the behavior of modular representations of numbers (much as in the spirit of the discovery of the quadratic reciprocity law). This is why I accepted to give a talk at this congress, which gathers some of the main experts in computing with residue number systems. Thanks again to the organizers for their invitation: I’m greatly looking forward to the conference!