Marek Wolf

Marek Wolf

"... upon looking at prime numbers one has
the feeling of being in the presence of one
of the inexplicable secrets of creation."

D.Zagier in Math. Intellig. 0 (1977), p.8, left column



I am a physicist at the Faculty of Mathematics and Natural Sciences.College of Sciences . In the late seventies I started working on strings and supersymmetry. This was my major field of interest until 1984, when I bought the famous ZX Spectrum computer (for less than 200$ !!!). Only then did I understand what good physics was. Immediately I turned from quantum field theory to fractals and chaos. Since Summer 1995 I am also interested in prime numbers, but I still believe that:

Physics is Fun !

e-mail: m.wolf at uksw.edu.pl

Curriculum vitae    (for pdf click here)    (look also here )

List of publications

Rare papers's


I am a member of the Editorial Board of the Computational Methods in Science and Technology and Waves, Wavelets and Fractals. Advanced Analysis




Below you can find my papers on the prime numbers:


This figure presents the main results reported in the paper:

Some Conjectures on the Gaps between Consecutive Primes (1.4 MB)

Here is gzip-ed file: Download "conjectures.ps.gz" (420 kB)



The more popular exposition of the above results is here:

Unexpected Regularities in the Distribution of Prime Numbers (840 kB)

Here is gzip-ed file: Download "primes.ps.gz" (260 kB)




In the paper "Generalized Brun's constants" it is argued, that the sum of reciprocals of all consecutive primes separated by distance d is equal to 4c_2/d \prod_{p\mid d} {p-1\over p-2}.

M.Wolf, Generalized Brun's constants (580 kB)

Here is gzip-ed file: "brun_gen.ps.gz" (180 kB)



In the paper "First Occurrence of a given gap between consecutive primes" the formula for the smallest prime such, that the next prime will be in the distance d is conjectured and compared with the results of the computer search:

First Occurrence of a given gap between consecutive primes (200kB)

Here is gzip-ed file: "firstocc.ps.gz" (68kB)

On the similar subject you can read the paper by Thomas Nicely.



In this paper the fractal structure on the set of prime numbers is described:

Download On the Twin and Cousin Primes (0.5 MB)

Here is gzip-ed file: "twins_ps.ps.gz" (110 kB)
Download fig.1b (770 kB)
Download fig.1c (1.3 MB)



In this paper the analog of the Skewes number for twins is considered:

An Analog of the Skewes Number for Twins (100 kB)



The   conjectures and results contained in the above papers are cited in the famous  Mathworld maintained by Eric Weisstein :
http://mathworld.wolfram.com/TwinPrimes.html
http://mathworld.wolfram.com/PrimeGaps.html
http://mathworld.wolfram.com/PrimeCountingFunction.html
http://mathworld.wolfram.com/PrimeDifferenceFunction.html
http://mathworld.wolfram.com/k-TupleConjecture.html
http://mathworld.wolfram.com/BrunsConstant.html
http://mathworld.wolfram.com/ShanksConjecture.html
http://mathworld.wolfram.com/CousinPrimes.html

There is a printed  version of this website CRC Concise Encyclopedia of Mathematics.

My papers are cited on the Wikipedia:
http://en.wikipedia.org/wiki/Brun's_constant
http://en.wikipedia.org/wiki/Cousin_prime
http://en.wikipedia.org/wiki/Riesz_function

My results are also cited at the famous  The On-Line Encyclopedia of Integer Sequences
maintained by N. J. A. Sloane:

http://oeis.org/A002496
http://oeis.org/A083844
http://oeis.org/A000101
http://oeis.org/A001912
http://oeis.org/A052291
http://oeis.org/A079296
http://oeis.org/A084255
http://oeis.org/A066081
http://oeis.org/A005250
and also here:
http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/surprising.htm
http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/NTfractality.htm
http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/isoc/evolutionnotes.htm
www.secamlocal.ex.ac.uk/people/staff/mrwatkin/isoc/fractality.htm
www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/1fnoisesigprocNT.htm
www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics6.htm
http://empslocal.ex.ac.uk/people/staff/mrwatkin/isoc/psizoom.htm


Here (server in USA) you can find the paper "Jumping Champions" (at this web site in Poland) written by Andrew Odlyzko,   Michael Rubinstein and me. It is about champions - the most often occuring gaps between consecutive primes. The most often occuring gaps are "primorials", i.e. products of consecutive primes. For example 6=2x3, 30=2x3x5, 210=2x3x5x7. If Dn denotes n-th champion Dn=2x3...xpn then they become most often occuring gap at N(n) which very roughly are given in the table below.

Because Andrew Odlyzko has the Erdos Number 1 , hence I have the Erdos Number 2. This paper was described by Ian Stewart in the December 2000 issue of the Scientific American on p.106.  Polish translation occured in Świat Nauki, luty 2001.


My papers are reviewed here by Matthew Watkins. I recommend his web page as a source of many very interesting articles about primes, zeta function etc.
I was astonished by the Riemann Series Theorem and I have written the program asking for a number and rearranging the alternating harmonic series to reach an accuracy epsilon and creating the Latex file with the output. By default the number is the Golden Ratio (approx 1,618033989) and epsilon is 0,001.

Here is a link to the weekly newspaper KURIER PLUS , where the interview Czy Bóg gra w koœci with me can be found. It is in Polish and deals with chaos, determinism, Plato, nature of mathematical theorems etc and was conducted by Elzbieta Kolakowska.