SOUS-ENSEMBLES et CONSTELLATIONS ARITHMETIQUES de NOMBRES PREMIERS
Résumé
Number theory (prime numbers) Since the Polignac 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 fairly regularly over the years ([2], [3], [4], [5], [6], [7], ...). For any e even greater than or equal to 2, we consider the sets C e of all pairs of primes (p, p+e), not necessarily consecutive. In the set P of prime numbers, we consider the subsets J e made up of all the distinct primes belonging to C e. We define the analogous sets C* e and J* e constituted only from all the pairs of primes (p, p+e) consecutive. The case e=2 defines the twin primes, e=4 the cousin primes, e=6 the sexy primes, e=8 the octo primes, ... Modular arithmetic in N/6N will allow us to establish two theorems concerning C e and C* e and expressing a new general property of these pairs. We will also give some numerical results on the rarities of J e and J* e using the calculation software PARI/GP. We will finally establish a theorem for constellations of prime numbers of k-tuple type, with constant deviation, where we will highlight the essential role of the primorial type deviations, e = p# (with p# = 2*3*5*7 *...*p). This result is the fairly natural extension, for k > 3, of the first two theorems which involve the smallest primorial greater than 2, 3# = 6. We will thus express simple constructive properties which explicitly determine the possible reasons of the arithmetic progressions of prime numbers of length k > 2, whose existence is ensured by the Green-Tao theorem ([5]). The reasons which lead, for a given k, to several progressions are only primorials strictly greater than or equal to 3# or their multiples and for k > 3 all the reasons are exclusively multiples of 6.
Domaines
Mathématiques [math]
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