Natural Wave Numbers, Natural Wave Co-numbers, and the Computation of the Primes

Abstract

The paper exploits an isomorphism between the natural numbers N and a space U of periodic sequences of the roots of unity in constructing a recursive procedure for representing and computing the prime numbers. The nth wave number ${\bf u}_n$ is the countable sequence of the nth roots of unity having frequencies k/n for all integer phases k. The space U is closed under a commutative and associative binary operation ${\bf u}_m \odot{\bf u}_n={\bf u}_{mn}$, termed the circular product, and is isomorphic with N under their respective product operators. Functions are defined on U that partition wave numbers into two complementary sequences, of which the co-number $ {\overset {\bf \ast }{ \bf u}}_n$ is a function of a wave number in which zeros replace its positive roots of unity. The recursive procedure $ {\overset {\bf \ast }{ \bf U}}_{N+1}= {\overset {\bf \ast }{ \bf U}}_{N}\odot{\overset {\bf \ast }{\bf u}}_{{N+1}}$ represents prime numbers explicitly in terms of preceding prime numbers, starting with $p_1=2$, and is shown never to terminate. If ${p}_1, ... , { p}_{N+1}$ are the first $N+1$ prime phases, then the phases in the range $p_{N+1} \leq k < p^2_{N+1}$ that are associated with the non-zero terms of $ {\overset {\bf \ast }{\bf U}}_{N}$ are, together with $ p_1, ...,p_N$, all of the prime phases less than $p^2_{N+1}$. When applied with all of the primes identified at the previous step, the recursive procedure identifies approximately $7^{2(N-1)}/(2(N-1)ln7)$ primes at each iteration for $ N>1$. When the phases of wave numbers are represented in modular arithmetic, the prime phases are representable in terms of sums of reciprocals of the initial set of prime phases and have a relation with the zeta-function.