A new approach to the research of the distribution of prime-triplets is developed. It differs from the usual methods (involving the sieve method) for this kind of research, and basing on Chebyshev inequality and on th...A new approach to the research of the distribution of prime-triplets is developed. It differs from the usual methods (involving the sieve method) for this kind of research, and basing on Chebyshev inequality and on the computation of average concentration of all the related subset. It leads to the proofs of following Lemma 2 and Theorem 2 (Lemma 1 and Theorem 1 in Reference 1 had been proved by means of this new method): Lemma 2 Among all the prime-triplet-subsets there exists at least one such subset which is an infinite set. Theorem 2 All the prime-triplet-subsets or infinitely many such subsets are infinite sets.Formulas for estimating the amount of such infinite sets are provided in this paper.展开更多
A new approach to the research into the distribution of prime pairs is developed, attached with corollaries about the distribution of prime triplets and generally about that of prime h-tuplets. It differs from the usu...A new approach to the research into the distribution of prime pairs is developed, attached with corollaries about the distribution of prime triplets and generally about that of prime h-tuplets. It differs from the usual methods(involving the sieve method) for this kind of research, and bases on Chebyshev inequality and on the computation of average concentration of all the related subsets, and leads to the proofs of following lemma and theorem, attached with four corollaries:[Lemma 1] Among all the prime-pair-subsets there exists at least one such subset (namely one of the sets of generalized prime twins) which is an infinite set.[Theorem 1] All the prime-pair-subsets (namely all sets of generalized prime twins) or infinitely many prime-pair-subsets are infinite sets.[Corollary 1] Among all the prime-triplet-subsets there exists at least one such subset which is an infinite set.[Corollary 2] All the prime-triplet-subsets or infinitely many prime-triplet-subsets are infinite sets.[Corollary 3] Among all the prime-h-tuplet-subsets there exists at least one such subset which is an infinite set, where h is an arbitrary finite integer ≥2.[Corollary 4] All the prime-h-tuplet-subsets or infinitely many prime-h-tuplet-subsets are infinite sets, where h is an arbitrary finite integer ≥2.展开更多
文摘A new approach to the research of the distribution of prime-triplets is developed. It differs from the usual methods (involving the sieve method) for this kind of research, and basing on Chebyshev inequality and on the computation of average concentration of all the related subset. It leads to the proofs of following Lemma 2 and Theorem 2 (Lemma 1 and Theorem 1 in Reference 1 had been proved by means of this new method): Lemma 2 Among all the prime-triplet-subsets there exists at least one such subset which is an infinite set. Theorem 2 All the prime-triplet-subsets or infinitely many such subsets are infinite sets.Formulas for estimating the amount of such infinite sets are provided in this paper.
文摘A new approach to the research into the distribution of prime pairs is developed, attached with corollaries about the distribution of prime triplets and generally about that of prime h-tuplets. It differs from the usual methods(involving the sieve method) for this kind of research, and bases on Chebyshev inequality and on the computation of average concentration of all the related subsets, and leads to the proofs of following lemma and theorem, attached with four corollaries:[Lemma 1] Among all the prime-pair-subsets there exists at least one such subset (namely one of the sets of generalized prime twins) which is an infinite set.[Theorem 1] All the prime-pair-subsets (namely all sets of generalized prime twins) or infinitely many prime-pair-subsets are infinite sets.[Corollary 1] Among all the prime-triplet-subsets there exists at least one such subset which is an infinite set.[Corollary 2] All the prime-triplet-subsets or infinitely many prime-triplet-subsets are infinite sets.[Corollary 3] Among all the prime-h-tuplet-subsets there exists at least one such subset which is an infinite set, where h is an arbitrary finite integer ≥2.[Corollary 4] All the prime-h-tuplet-subsets or infinitely many prime-h-tuplet-subsets are infinite sets, where h is an arbitrary finite integer ≥2.
