Abstract
<p>Euler{'}s formula establishes the relationship between the trigonometric function and the exponential function. In doing so unifies two waves, a real and an imaginary one, that propagate through the Complex number set, establishing relation between integer numbers. A complex wave, if anchored by zero and by a defined integer number \textit{N}, only can assume certain oscillation modes. The first mode of oscillation corresponds always to a \textit{N} prime number and the other modes to its multiples.</p><p>\begin{center}<br />\(\psi (x)=x e^{i\left(\frac{n \pi }{N}x\right)}\)<br />\end{center}</p><p>Under the above described conditions, these waves and their admissible oscillation modes allows for primality testing of integer numbers, the deduction of a new formula $\pi(x)$ for counting prime numbers and the identification of patterns in the prime numbers distribution with computing time gains in the calculations. In this article, four theorems and one rule of factorizing are put forward with consequences for prime number signaling, counting and distribution. Furthermore, it is establish the relationship between this complex wave with a time independent semi-classical harmonic oscillator, in which the spectrum of the allowed energy levels are always only prime numbers. Thus, it is affirmative the reply to the question if the prime numbers distribution is related to the energy levels of a physical system.</p>
Highlights
For Mathematics, numbers are dimensionless and appear to be abstract, it seems that there are some rules guiding the construction of these abstractions
The relationship of the Riemann Zeta with Prime Numbers (PNs) is well known and, in this work it is established the relationship of PNs with the spectrum of energy levels allowed in a semi-classical oscillator with hermitian operator
In doing so establishes relationships between an integer number and its divisors and multiples. This approach allows to establish six theorems that determine whether an integer number is a PN, if a range of integer numbers contains PNs and where these prime numbers can occur, providing the foundation for a new formula to count PNs
Summary
For Mathematics, numbers are dimensionless and appear to be abstract, it seems that there are some rules guiding the construction of these abstractions. The overwhelming majority of these formulas is considered useless from the practical point of view, each with more or less theoretical interest Some of these formulas allow counting primes or help establish limits for the π(x) function. Barry and Keating speculate that the Riemann zeta function dynamics is related with the semiclassical hamiltonian Hcl = xp (Berry and Keating, 1999) This approach has been developed by several authors, Sierra-Laguna or Menezes-Svaiter for example, where it is search a physics solution for the PNs and Riemann hypothesis (Sierra, 2010, 2008; Gutzwiller, 1990; Sierra and Rodriguez-Laguna, 2011). The pre-conditions established and its consequences are discussed
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