Representation and Generation of Prime and Coprime Numbers by Using Structured Algebraic Sums

The algebraic structure and distribution of prime numbers remain two of the most fundamental problems in mathematics. The Fundamental Theorem of Arithmetic, proved by Euclid, and Goldbach’s conjecture, while universal in scope with respect to how numbers can be represented multiplicatively or additively, do not provide insights into the structure of primes. Similarly, the definition of a prime −as a number divisible only by 1 and itself− or a sieve algorithm, commonly used to generate primes by successively eliminating multiples, offer no insight into the structure of primes. The powerful and persistent consideration of prime numbers as universal “arithmetic quanta” has not necessitated an equally powerful need for parallel research into a deeper and possibly more insightful explanation of primeness, that is, a better understanding of “why” a number is prime. In this paper, prime and coprime numbers are represented and generated by algebraic expressions. Specifically, given the first <i>n</i> primes, <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>, sufficient conditions are given for expressing primes greater than <i>p<sub>n</sub></i>, and coprimes with prime factors greater than <i>p<sub>n</sub></i>, as algebraic functions of <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>. Thus, primality and co-primality are shown to be mathematical properties with inherently evolutionary algebraic characteristics, since larger primes and coprimes can be generated algebraically from smaller ones. The methodology described in the paper can be a useful tool in the study and analysis of the complexity, structure, interrelationships and distribution of primes and coprimes.