Version 1
: Received: 23 November 2023 / Approved: 27 November 2023 / Online: 27 November 2023 (11:09:06 CET)
How to cite:
Orús-Lacort, M.; Orús, R.; Jouis, C. Analyzing Twin Primes, Goldbach’s Strong Conjecture and Polignac’s Conjecture. Preprints2023, 2023111660. https://doi.org/10.20944/preprints202311.1660.v1
Orús-Lacort, M.; Orús, R.; Jouis, C. Analyzing Twin Primes, Goldbach’s Strong Conjecture and Polignac’s Conjecture. Preprints 2023, 2023111660. https://doi.org/10.20944/preprints202311.1660.v1
Orús-Lacort, M.; Orús, R.; Jouis, C. Analyzing Twin Primes, Goldbach’s Strong Conjecture and Polignac’s Conjecture. Preprints2023, 2023111660. https://doi.org/10.20944/preprints202311.1660.v1
APA Style
Orús-Lacort, M., Orús, R., & Jouis, C. (2023). Analyzing Twin Primes, Goldbach’s Strong Conjecture and Polignac’s Conjecture. Preprints. https://doi.org/10.20944/preprints202311.1660.v1
Chicago/Turabian Style
Orús-Lacort, M., Román Orús and Christophe Jouis. 2023 "Analyzing Twin Primes, Goldbach’s Strong Conjecture and Polignac’s Conjecture" Preprints. https://doi.org/10.20944/preprints202311.1660.v1
Abstract
Here we analyze three well-known conjectures: (i) the existence of infinitely-many twin primes, (ii) Goldbach’s strong conjecture, and (iii) Polignac’s conjecture. We show that the three conjectures are related to each other. In particular, we see that in analysing the validity of the Goldbach’s strong conjecture, one must consider also the existence of an infinite number of twin primes. As a consequence of how we approach this analysis, we also observe that if this conjecture is true, then so is Polignac’s conjecture. Our first step is an analysis of the existence of infinitely-many twin primes numbers. For this, using the formula 4((n−1)!+1) ≡ −n (mod n(n + 2)) – satisfied if and only if (n, n + 2) are twin primes –, together with Wilson’s theorem, we obtain conditions that must be met for two numbers to be twin primes. Our results, obtained from an analytic and functional study, lead us to conclude that there may exist infinitely-many twin primes. Next, we consider the validity of the Goldbach’s strong conjecture. After showing that the conjecture is true for the first even numbers, we notice a pattern that we analyze for any even number, reducing it to three cases: (i) when the even number 2n is two times a prime number n; (ii) when the even number 2n is such that n = 2m, with 2m − 1 and 2m + 1 twin primes; (iii) all other cases, i.e., for any 2n even number ∀n ∈ N with n > 1, n prime or not, with independence of n = 2m being 2m−1 and 2m+1 twin primes or not. In this last case we show that one can always find a certain r ∈ N such that 1 < r < n satisfying that n−r and n+r are primes, so that their sum is 2n. In this case we use the reduction to absurd method, and our results lead us to conclude that Goldbach’s strong conjecture is true to the best of our calculations, as well as the Polignac’s conjecture.
Keywords
Number Theory; Twin Primes; Goldbach’s Strong Conjecture; Polignac’s Conjecture
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
