Towards a generic view of primality through multiset decompositions of natural numbers

Abstract

Factorization results in multisets of primes and this mapping can be turned into a bijection between multisets of natural numbers and natural numbers. At the same time, simpler and more efficient bijections exist that share some interesting properties with the bijection derived from factorization.This paper describes mechanisms to emulate properties of prime numbers through isomorphisms connecting them to computationally simpler representations involving bijections from natural numbers to multisets of natural numbers.As a result, interesting automorphisms of N and emulations of the rad, Möbius and Mertens functions emerge in the world of our much simpler multiset representations.Finally we generalize the multiset decomposition mechanism derived from the Diophantine equation 2x(2y+1)=z and study emulations of the behavior of the ω function that counts the number of distinct prime factors of a number as well as the equivalent of b-smoothness (a property characterizing natural numbers having all their prime factors smaller than b).The paper is organized as a self-contained literate Haskell program. The code extracted from the paper is available as a standalone program at http://logic.cse.unt.edu/tarau/research/2012/jprimes.hs.