CONSTELLATIONS QUELCONQUES DE NOMBRES PREMIERS
Résumé
Number theory (prime numbers) ANY CONSTELLATIONS OF PRIME NUMBERS. Following Polignac's conjecture ([1]) stated in 1849, a large number of works on prime numbers, pairs of primes distant by an even integer and constellations of primes have been published ([2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]), with, in particular, Brun's theorem ([2]) in 1919 and Green-Tao ([7]) in 2008. This last theorem demonstrates the existence of arithmetic constellations of length n > 2 and ratio e, for all n; in ([8]) we have established properties allowing us to explicitly determine the possible reasons e of these arithmetic sequences and to exhibit numerous representatives of them. We are considering here constellations where the successive increasing distinct prime numbers of the sequence are distant by any gaps, and there is, a priori, no constraint imposed on these gaps which are therefore independent of each other; we will speak of free or arbitrary constellations (or n-tuples). Using modular arithmetic in N/6N, we will establish constructive properties (as in [12] for specifically arithmetic constellations) to determine representatives of various free n-tuples for n > 2. Theorems on free triples, quadruplets and quintuplets will be established and will perfectly allow to exhibit various representatives of these n-tuples according to the modulo 6 properties of the gaps between the first constituents. For all free n-tuples, n > 2, we will show that for some modulo 6 properties of their gaps, there will be several representatives that we can exhibit, but naturally, all n-tuples with arbitrary gaps cannot not exist, as do all arithmetic tuples for whatever reason. By admitting Polignac's conjecture, we can expand the Green-Tao theorem ([7]) and state that, among prime numbers, there always exist n-tuples, free or not, for all n. For any free n-tuple, the equality modulo 6 of all its gaps will ensure a large number of representatives, as a ratio e equal to a certain primorial (depending on n) did for any arithmetic ntuple ([12]). In the context of data protection, we finally propose an application of these results to build one-way functions that produce a secret code, in the form of a prime number, associated with digital personal data that could thus be found protected. Thanks to the existence and possible effective construction of very many representatives of free n-tuples (n > 2) it is possible to manufacture and use a very large number of such functions which can therefore be perfectly personalized; a simple example is provided in an appendix link. The proposed numerical illustrations were obtained using the PARI/GP calculation software.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)