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Turku Symposium on Number Theory in Memory of Kustaa Inkeri
May 31 - June 4, 1999
University of Turku
Turku, Finland |
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Organizers
Matti Jutila, Tauno Metsänkylä
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On the Twin and Cousin Primes
by
Marek Wolf
Institute of Theoretical Physics, University of Wroclaw, PL-50-204 Wroclaw, Pl. M. Borna 9
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On the Twin and Cousin Primes |
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Institute of Theoretical Physics, University of Wrocaw |
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Pl.Maxa Borna 9, PL-50-204 Wrocaw, Poland |
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e-mail: mwolf@ift.uni.wroc.pl |
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 new formula expressing the number \pi2(x) of twins smaller
than x directly by \pi(x) - the total number of primes up to
x - of
the form \pi2(x) \approx C2 \pi2(x)/x is proposed and compared with the
computer search.
The plot of the function W(x)=\pi2(x)-\pi4(x) giving the
difference of the number of twins and cousins for x in (1, 1012) 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 with the fractal dimension 0.51. It is conjectured that the
number of zeros o
the function W(x) up to x is roughly \surd{x/\pi}, here
\pi = 3.14 ... . 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 also equal to 1.48. It
is conjectured that the maximal gap between a pair of
consecutive twins < x grows like log3(x).
http://www.ift.uni.wroc.pl/~mwolf
Date received: April 14, 1999
Copyright © 1999 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Conferences Inc.
Document # cacf-32.