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First Occurrence of a Given Gap between Consecutive Primes
, 1997
"... Heuristic arguments are given, that the pair of consecutive primes separated by a distance d appears for the first time at p f (d) ¸ p d exp ` 1 2 q ln 2 (d) + 4d ' . The comparison with the results of the computer search provides the support for the conjectured formula. ..."
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Heuristic arguments are given, that the pair of consecutive primes separated by a distance d appears for the first time at p f (d) ¸ p d exp ` 1 2 q ln 2 (d) + 4d ' . The comparison with the results of the computer search provides the support for the conjectured formula.
Characterization of the distribution of twin primes
 In: arXiv:math.NT/0103191
"... We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like fixed probability random events. As the sequence of primes grows ..."
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We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like fixed probability random events. As the sequence of primes grows, the probability decreases as the reciprocal of the count of primes to that point. The manner of the decrease is consistent with the Hardy–Littlewood Conjecture, the Prime Number Theorem, and the Twin Prime Conjecture. Furthermore, our probabilistic model, is simply parameterized. We discuss a simple test which indicates the consistency of the model extrapolated outside of the range in which it was constructed.
Implications of a New Characterisation of the Distribution of Twin Primes
, 2001
"... We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the “high jumper ” i.e., the separat ..."
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We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the “high jumper ” i.e., the separation with greatest likelihood (in terms of primes) is always expected to be zero.
Generalized Brun's constants
, 1997
"... It is argued, that the sums of reciprocals of all consecutive primes separated by gaps of the length d for d 6 are equal to 4c 2 d Q pjd;p?2 p\Gamma1 p\Gamma2 , where c 2 = Q p?2 i 1 \Gamma 1 (p\Gamma1) 2 j . Besides some heuristic arguments leading to this formula, there is also compa ..."
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It is argued, that the sums of reciprocals of all consecutive primes separated by gaps of the length d for d 6 are equal to 4c 2 d Q pjd;p?2 p\Gamma1 p\Gamma2 , where c 2 = Q p?2 i 1 \Gamma 1 (p\Gamma1) 2 j . Besides some heuristic arguments leading to this formula, there is also comparison with the results of computer investigation up to 2 42 ß 4:4 \Theta 10 12 . These "experimental" data provides supports for the conjectured formulae. It is also shown, how the guessed formula reproduces the well known fact, that the sum of reciprocals of all primes p ! x grows like ln(ln(x)). 2 Marek Wolf 1. In 1919 Brun [1] has shown that the sum of the reciprocals of all twin primes is finite: B 2 = ` 1 3 + 1 5 ' + ` 1 5 + 1 7 ' + ` 1 11 + 1 13 ' + : : : ! 1: (1) Sometimes 5 is included only once, but here I will adopt the above convention. The analytical formula for B 2 is unknown 1 and the sum (1) is called the Brun constant [2]. The numerical estimations give ...
DRAFT KRMK Implications of a New Characterisation of the Distribution of Twin Primes
, 2001
"... We bring to bear an empirical model of the distribution of twin primes and produce three distinct results pertinent to twins and, by extension, evidence against the Riemann Hypothesis. The first result is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the seque ..."
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We bring to bear an empirical model of the distribution of twin primes and produce three distinct results pertinent to twins and, by extension, evidence against the Riemann Hypothesis. The first result is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the “high jumper ” i.e., the separation with greatest likelihood (in terms of primes) is always expected to be zero. The third is an elementary proof that Brun’s constant is bounded, i.e., that the series of reciprocal twins converges. We will demonstrate that our elementary proof is necessarily flawed because it is too strong, and attribute its failure to the fact that the error term was neglected in the model for the distribution of the primes. It is made very clear that the proof is incorrect by the fact that it is easily adapted to demonstrate that sums of subsequences of reciprocal primes are bounded, whereas it is clear that all series of the type we consider are in fact divergent. Attempts to explicitly model the behaviour of the error term require consideration of the
Some Remarks on the Distribution of twin Primes
, 2001
"... The computer data up to 2 44 ≈ 1.76×10 13 on the gaps between consecutive twins is presented. The simple derivation of the heuristic formula describing computer results contained in the recent papers by P.F.Kelly and T.Pilling [5], [6] is provided and compared with the “experimental ” values. Key wo ..."
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The computer data up to 2 44 ≈ 1.76×10 13 on the gaps between consecutive twins is presented. The simple derivation of the heuristic formula describing computer results contained in the recent papers by P.F.Kelly and T.Pilling [5], [6] is provided and compared with the “experimental ” values. Key words: Prime numbers,twins MSC: 11A41 (Primary), 11Y11 (Secondary) 2 Marek Wolf Among the primes the subset of twin primes is distinguished: twins are such numbers (p, p ′ ) that both p and p ′ = p+2 are prime. So the set of twins starts with (3, 5), (5, 7), (11, 13), (17, 19), (29, 31).... It is not known whether there is infinity of twins; the largest known today pair of twins was found recently by Underbakke, Carmody, Gallot (see