Argument for a twin primes theorem. Landscapes, panoramas and horizons within Eratosthenes sieve. Hubert Schaetzel To cite this version: Hubert Schaetzel. Argument for a twin primes theorem. Landscapes, panoramas and horizons within Eratosthenes sieve.. 2020. hal-02945018 HAL Id: hal-02945018 https://hal.archives-ouvertes.fr/hal-02945018 Preprint submitted on 21 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Number Theory / Théorie des nombres Argument for a twin primes theorem. Landscapes, panoramas and horizons within Eratosthenes sieve. Hubert Schaetzel Abstract We explore three ways on the twin primes problem. We start with the intermediate sets produced by Eratosthenes sieve implementation. Properties related to the proportions of integers eliminated during process on one hand and the distances generated between integers on the other hand allow twice deducing the infinity of prime numbers and twin prime numbers. In the former case, the analysis of the proportions also allows getting an asymptotic evaluation similar to Hardy-Littlewood formula, but without fully valid proof. In the latter case, the analysis of the spacings between remaining integers yields a replica of Bertrand’s postulate with approximate 2pi spacing and the asymptotic evaluation of the maximum of the distances between pairs of numbers (of spacing 2), which is ranging around ∑i 2pk, enables to conclude to 2 the divergence of twin prime numbers below abscissa pi . Finally, an alternative method, that is readily generalizable to many Diophantine equations, is proposed as an invitation to new studies. Again, we infer the Euler product suggested by Hardy-Littlewood. Argumentaire pour un théorème des nombres premiers jumeaux. Paysages, panoramas et horizons du crible d’Eratosthène. Résumé Nous étudions trois approches au problème des premiers jumeaux. Nous commençons par les ensembles intermédiaires produits par l’exécution du crible d’Eratosthène. Les propriétés liées aux proportions de nombres entiers éliminés d’une part et aux espacements générés entre nombres entiers d’autre part permettent par deux fois de déduire l’infinité des nombres premiers, puis des nombres premiers jumeaux. Dans le premier cas, l’analyse des proportions permet également d’obtenir une évaluation asymptotique identique à la formule d’Hardy-Littlewood, mais sans pleine et entière démonstration. Dans le second cas, l’analyse des espacements entre nombres restants permet d’obtenir une réplique du Postulat de Bertrand avec un espacement de l’ordre de 2pi et l’évaluation asymptotique du maximum de la distance entre paires de nombres (d’écart 2), évaluation qui est équivalente à ∑i 2pk, permet de conclure à la divergence du 2 cardinal des nombres premiers jumeaux en dessous de l’abscisse pi . Enfin, une méthode alternative aisément généralisable à de nombreuses équations diophantines est proposée en guise d’invitation à d’autres études. Nous en déduisons à nouveau le produit d’Euler suggéré par Hardy-Littlewood. Status Preprint Date Version 1 : November 29, 2011 Version 17 : September 11, 2020 Paradox see [1] - no prime number is even except one - no prime number is even except two P 1/142 Summary 1. Preamble. 3 2. An expeditious proof. 3 3. Terminology. 3 4. Fundamental theorems. 4 4.1. Three theorems. 4 4.2. Generalization of Mertens theorem. 4 4.3. Logarithm weighted sums. 5 5. Eratosthenes sieve. 6 5.1. Depletion algorithm. 6 5.2. Landscaping of spacings between pseudo primes. 13 5.2.1. Panoramas of populations. 13 5.2.2. Horizons on the iterative enumeration of populations. 15 5.2.3. Evolution of aggregated populations. 22 5.2.4. Cradle of the multiplicative factors. 23 5.2.5. Maximal spacing. 39 5.2.6. Minimal spacing. 42 6. Eratosthenes crossed sieve. 43 6.1. Case of the twin prime numbers. 43 6.2. Case of relative prime numbers. 46 6.3. Evaluation of relative prime numbers cardinals. 49 6.3.1. Case of twin prime numbers. 49 6.3.2. Case of pairs of prime numbers distant of 2m. 50 6.3.3. Case of relative prime numbers. 51 6.3.4. Formula of cardinals. 51 6.3.5. Common asymptotic branches. 52 6.3.6. Implementation of a bijection between relative prime numbers with common asymptotic branches. 53 6.3.7. Hardy-Littlewood formula. 53 6.3.8. Comparative evolution of depletion coefficients. 56 6.4. Landscaping of twin numbers spacings. 57 6.4.1. Generalities. 57 6.4.2. Basic idea. 58 6.4.3. Panoramas od enumeration. 58 6.4.4. Generative process. 63 6.4.5. Extrema research. 66 6.4.6. Algorithmic background. 68 6.4.7. Classes. 70 6.4.8. Configurations. 71 6.4.9. Spacings generated by the sieve. 72 6.4.10. Lower and upper bounds. 77 6.4.11. Futher horizons for spacings. Entities viewed with a telescope. 79 6.5. Landscaping of spacings between relative integers. 82 6.5.1. Periodicity of the entities. 86 4.4.11.1 Periodicity focusing on even components. 86 4.4.11.2 Periodicity focusing on odd components. 89 6.5.2. Sums of products. 92 6.5.3. Divergence of solutions. 94 6.6. Comparison of families. 94 7. Theorem of density of prime numbers. 96 7.1. Equivalent of a prime number variable. 96 7.2. Reconstruction of De Polignac formula. 97 P 2/142 1. Preamble. Rarefaction of pairs of primes distant of a given value 2n is a simple process based on Eratosthenes sieve. We will establish our theorems thanks to arithmetical laws governing integers’ depletions within natural numbers set N while implementing the said algorithm. The article below has certainly nothing complicated for specialists of this topic. To give more clarity and strength to the argument, we will apply it initially to the enumeration of the prime numbers, i.e. we will attempt to retrieve the prime number theorem (PNT). Passing from the prime numbers’ case to the twin prime numbers will simply consist of replacing a given law of scarcity th (pi-1)/pi by another (pi-2)/pi, pi being the i odd prime number. Although the Hardy-Littlewood formula is deduced, no proof is given (nor for the PNT). Only, the infinitely of twin prime numbers is deduced (following similar work on prime numbers). We downgrade and overrate the number of solutions in both cases and the resulting framing show upper and lower boundaries converging asymptotically (and tending towards infinity). The study is also upgraded after that for our two groups of objects, prime numbers and twin prime numbers, by evaluating the distances between elements, the guiding threads being now, more or less, the two expressions 2pi and ∑i 2pk respectively. However, more than the results in the pi to pi² range that allow us to conclude upon the stated problem, we focus attention, when running Eratosthenes algorithm, on the existence of recursive formulas’ systems to evaluate asymptotically in the pi to pi+pi# interval, pi# = 2.3.5.7.11…pi denoting the primorial of pi, the integers’ populations with given spacing Δ = 2j (populations of pseudo-primes on the one hand, populations of pseudo-twin primes or relative primes on the other hand), knowing less than j/2+1 initial staffs. (The terms “pseudo” and “spacing” will be defined very soon in the present article). Thus, the interest of this article has also become over the course of the different versions, this aspect having taken more and more importance with respect to the initial purpose, facing apparent absence of such a corpus elsewhere, that of an in- depth study of the Eratosthenes sieve. The reader would have been disappointed with the lack of challenge if he had already found here all the statements demonstrated. On the contrary, and fortunately, he will still be able to exercise all of his insight facing high walls of difficulties, especially in order to appropriate himself the said recursive systems. There is a time for discovery and another for the full comprehension of a subject. 2. An expeditious proof. For the reader who does not have time, here is an appetizier for his immediate satisfaction. Proposition 1 There is an infinity of twin prime numbers. Proof Let us apply the Eratosthenes algorithm up to step pi. Then, beyond pi, the intervals of size #pi, the primordial of pi, contain each ∏ (pi-2) pairs of 2-gap numbers. This answers the question of the existence of pairs (not necessarly primes). As the algorithm begins with the removal of the smallest dividers, the first pair is a pair of twin primes (you can challenge anyone to find a counter-example). Let us consider pj the largest number of this pair. Let us continue the depletion algorithm up to pj. Beyond pj, the intervals of size #pj each contain ∏ (pj-2) pairs of 2-gap numbers, the first of which is a pair of twin prime numbers which is different of the first pair. So we get a second coveted pair. The argument applies to infinity by recurrence. 3. Terminology. Gap and spacing : Notions related to the distance between objects in this study can lead to pernicious confusion. Precise terminology is therefore required to avoid it. We will have to manipulate either isolated integers or pairs of integers.