THE GREATEST PRIME FACTOR AND RELATED MAGMAS

Mihai Caragiu (U.S.A.) and Greg Back (U.S.A.)

Received November 21, 2009

Abstract

By using the greatest prime factor function, we introduce a family of infinite non-associative magmas on the set of prime numbers, and explore the properties of one such particular commutative magma P which incorporates the greatest prime factor function in conjunction with the prime addition, thus expressing the rich structure at the interface between the additive and the multiplicative properties of integers. An investigation of the set of finite submagmas of P reveals interesting connections with properties of primes, and shows that every non-singleton submagma of P contains a certain four-element submagma.

Keywords and phrases: magmas, primes, greatest prime factor.