@MISC{Wolf96onthe, author = {Marek Wolf and Marek Wolf}, title = {On the Twin and Cousin Primes}, year = {1996} }

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Abstract

The computer results of the investigation of the number of pairs of primes separated by gap d = 2 ("twins") and gap d = 4 ("cousins") are reported. The number of twins and cousins turn out to be is almost the same. The plot of the function W (x) giving the difference of the number of twins and cousins for x 2 (1; 10 12 ) is presented . This function has fractal properties and the fractal dimension is approximately 1.48 --- what is very close to the fractal dimension of the usual Brownian motion. The set of primes, up to which the numbers of twins and cousins are exactly the same seems to have the fractal structure. The box-counting method gives the fractal dimension of this set approximately 0.51. The statistics of distances between primes being the zeros of W (x) display the cross-over from the exponential decrease to the power like dependence with the exponent equal 1.48. Arguments that W (x) has the same properties as a typical sample path of the random walk are given. The analog of the Brun's constant is defined for cousins. 2 Marek Wolf 1. In the paper [1] Hardy and Littlewood have proposed about 15 conjectures. The conjecture B of their paper states 1 : There are infinitely many primes pairs p; p 0 = p + d; (1) for every even d. If d (x) is the number of pairs less than x, then d (x) c 2 x ln 2 (x) Y pjd p \Gamma 1 p \Gamma 2 : (2) Here the constant c 2 (sometimes called "twin--prime " constant, see [3]) is defined in the following way: c 2 j 2 Y p?2 ` 1 \Gamma 1 (p \Gamma 1) 2 ' = 1:32032 . . . (3) Nobody has proved as yet (2), even there is no proof that there is infinity of twin (d = 2) primes. The largest twins known officially are: 697053813 \Theta 2 16352 \Sigma 1 (4) found recently by Indlekofer and Jarai, [2]. The pairs of ...