Prime and Quasi-Prime Number Races

Abstract

We review the body of work done on prime number races, specifically the results involving infinitely many lead changes in prime number races. We describe a computational way of showing that any race has infinitely many lead changes and greatly expand the known results in this area. An extension of the traditional prime number race problem is discussed where we race "quasi-primes" or composite numbers that are the product of two odd primes modulo 4. We then consider what "percentage" of the time that the residue class 1 leads the residue class 3 in this "quasi-prime" race modulo 4.