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2019•Computational Methods in Science and TechnologyOpen access

On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood

László Fejes Tóth

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Abstract

In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples.In 2011, Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs.In this paper, we find "Skewes numbers" for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture.Moreover, we present several algorithms to compute such numbers.

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In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples.In 2011, Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs.In this paper, we find "Skewes numbers" for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture.Moreover, we present several algorithms to compute such numbers.

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Available abstract

In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples.In 2011, Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs.In this paper, we find "Skewes numbers" for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture.Moreover, we present several algorithms to compute such numbers.

Key concepts: Tuple, Twin prime, Conjecture, Prime (order theory), Mathematics, Combinatorics, Prime number, Natural density

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