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The Greatest Prime Factor and Related Magmas (journal paper)

Mihai Caragiu and Greg Back
Ohio Northern University

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. (abstract available at )

JP Journal of Algebra, Number Theory and Applications, 15(2), 127-136 (December 2009)