Ordinals of paired prime numbers as addends (an algorithm) Ordinales de números primos pareados como sumandos (un algoritmo)

Abstract

This essay is situated within the field of additive number theory, with particular emphasis on problems concerning the representation of integers as sums of prime numbers.

The scope of this article is (dissemination/education) and it does not present new proven results, but rather a framework for exploration/recording to engage the reader with the conjectures that are stated. Through a simple "pairing" algorithm of sums that result in even numbers from some of the elements of the "source" set, which begins at 8 and progresses according to k+4 ad infinitum: A = {8+4k: k ∈ ℕ₀} = {8, 12, 16, 20, 24, ...}; it aims to illustrate to the general public how the "Strong (binary) Goldbach Conjecture" and the "Conjecture on the Infinitude of Twin Primes" relate to each other. With the proposed algorithm, a "Recording Space" is also defined, which is an ever-growing "zone" where the ordinals of the "first pairs" of prime summands (twin or not) are recorded for each element of the set A = {8+4k: k ∈ ℕ₀}, as defined above. It is assumed that the ordinal for twin primes is the expression "1". Finally, it is conjectured that the relationship between twin primes and the numbers known as Fermat's "Pythagorean primes" of the form 4k+1 (Fermat's Theorem on the Sum of Two Squares) illustrates, in turn, the relationship between the "Strong (binary) Goldbach Conjecture" and the "Conjecture on the Infinitude of the Set of Twin Primes".