NOMBRES PREMIERS DE SOPHIE GERMAIN et VARIANTES
Résumé
Number theory (prime numbers) PRIME NUMBERS OF SOPHIE GERMAIN and VARIANTS. We are interested here in the prime numbers of Sophie Germain (prime p such that 2p+1 is also prime), not as the first stone of the long process (SG) towards the resolution of the Fermat conjecture, which led to the Fermat-Wiles theorem ([9]) in 1994, but as a certain set of prime numbers, denoted PSG (strictly included in the set P of all primes), of which we will study, as for P ([4], [7], [15], [16], ...), couples (twins, cousins, sexy, ...), triplets or other arithmetic tuples or not. This analysis will be made (as in [15], [16]) within the framework of the modular arithmetic of Gauss, with always the essential role played by the classes modulo 6 of N/6N. In PSG, we will show, in particular, that there are only for a gap e = 0(6) that there exist several pairs and that there can exist several arithmetic triples; for e not 0(6), there is no triple or pair, except possibly (if 3+e PSG), the unique pair (3, 3+e) for e = 2(6).< br/> We will also consider ∈ other sets of prime numbers of the same type (inspired by the generalized Cunningham chains ([3), [5])) (AC), by swapping the 2p+1 bond of PSG for ap+b, with a and b relatively prime; a property of the prime number 3 then one of the odd integer 3 will result. For the sets PSG, G 2-1 (defined by 2p-1) and G4 (defined by 2p+4), with a deviation e not 0(6), there is at best in some cases only one possible pair but never an arithmetic triple and it takes e = 0(6) for several representatives to be possible; for a and b not 0(6), there exists at best any triple starting with p = 3. G6 (defined by 6p+1) is much richer and comes close to the situation in P ([15], [16 ]), with at least one representative (case of couples and arithmetic triples) for almost each value of e. We can list online prime numbers belonging to the sets PSG, G 2-1 , G4 and G6.
Domaines
Mathématiques [math]
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