A Heuristic Asymptotic Formula Concerning the Distribution of Prime Numbers

Bateman, Paul T.; Horn, Roger A.

doi:10.2307/2004056

Cited by 105 publications

(240 citation statements)

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“…The predictions were surprisingly accurate and further study indicated that our data could be used to support a conjecture of Bateman and Horn [1] which in turn supported the famous "Hypothesis H" of Schinzel and Sierpiński [11].…”

Section: Introductionsupporting

confidence: 75%

Distribution of generalized Fermat prime numbers

Dubner¹,

Gallot²

2001

Math. Comp.

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Abstract. Numbers of the form F b,n = b 2 n +1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2 m −1. The theoretical distributions of GFN primes, for fixed n, are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Bateman and Horn's quantitative form of "Hypothesis H" of Schinzel and Sierpiński. A list of the current largest known GFN primes is included.

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Section: Introductionsupporting

confidence: 75%

Distribution of generalized Fermat prime numbers

Dubner¹,

Gallot²

2001

Math. Comp.

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“…Hardy and Littlewood [4] conjectured that the number of primes p S N such that p + 2r is also prime is is the "twin-prime" constant, and q runs over the odd primes (this convention is adopted throughout). Conjecture (1) has been substantiated empirically and extended by several authors, e.g. [1], [3], [5] and [6].…”

Section: Introductionmentioning

confidence: 90%

“…Conjecture (1) has been substantiated empirically and extended by several authors, e.g. [1], [3], [5] and [6].…”

Section: Introductionmentioning

confidence: 90%

The Distribution of Small Gaps Between Successive Primes

Brent¹

1974

Mathematics of Computation

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Abstract. For r jg 1 and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes p Ú N such that p + 2r is the least prime greater than p is asymptotic to f (á"oo7^) dx> where the Ar,k are certain constants. We describe a method for computing these constants. Related constants are given to 10D for r = 1(1)40, and some empirical evidence supporting the conjecture is mentioned.

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“…Furthermore, a conjecture of Bateman and Horn [9] vastly generalizes the prime number theorem to give the expected density of such prime values. For our purposes we must also consider polynomials with rational coefficients; our definition incorporates the natural generalization of these conjectures to such polynomials.…”

Section: 1mentioning

confidence: 99%

A Taxonomy of Pairing-Friendly Elliptic Curves

2009

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Abstract. Elliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairingbased cryptographic systems. Such "pairing-friendly" curves are rare and thus require specific constructions. In this paper we give a single coherent framework that encompasses all of the constructions of pairing-friendly elliptic curves currently existing in the literature. We also include new constructions of pairing-friendly curves that improve on the previously known constructions for certain embedding degrees. Finally, for all embedding degrees up to 50, we provide recommendations as to which pairing-friendly curves to choose to best satisfy a variety of performance and security requirements.

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A Heuristic Asymptotic Formula Concerning the Distribution of Prime Numbers

Cited by 105 publications

References 1 publication

Distribution of generalized Fermat prime numbers

Distribution of generalized Fermat prime numbers

The Distribution of Small Gaps Between Successive Primes

A Taxonomy of Pairing-Friendly Elliptic Curves

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