Computer Experiments with Mersenne Primes
Cardinal Stefan Wyszynski University
Faculty of Mathematics and Natural Sciences, College of Sciences
ul. Wóycickiego 1/3, Auditorium Maximum, (room 113)
PL-01-938 Warsaw, Poland
E-mail: m.wolf@uksw.edu.pl
Received:
Received: 14 March 2013; revised: 1 July 2013; accepted: 17 July 2013; published online: 5 September 2013
DOI: 10.12921/cmst.2013.19.03.157-165
OAI: oai:lib.psnc.pl:451
Abstract:
We have calculated on the computer the sum BM of reciprocals of first 47 known Mersenne primes with the accuracy of over 12000000 decimal digits. Next we developed BM into the continued fraction and calculated geometrical
means of the partial denominators of the continued fraction expansion of BM . We get values converging to the Khinchin’s constant. Next we calculated the n-th square roots of the denominators of the n-th convergents of these continued fractions obtaining values approaching the Khinchin-Lèvy constant. These two results suggests that the sum of reciprocals of all Mersenne primes is irrational, supporting the common belief that there is an infinity of the Mersenne primes. For comparison we have done the same procedures with a slightly modified set of 47 numbers obtaining quite different results. Next we investigated the continued fraction whose partial quotients are Mersenne primes and we argue that it should be transcendental.
Key words:
References:
[1] B. Adamczewski and Y. Bugeaud, On the Maillet-Baker
continued fractions. Journal für die reine und angewandte
Mathematik, 606, 105-121 (2007).
[2] B. Adamczewski and Y. Bugeaud, A short proof of the tran-
scendence of thue-morse continued fractions. American Math-
ematical Monthly 114, 536-540 (2007).
[3] B. Adamczewski, Y. Bugeaud, and L. Davison, Continued
fractions and transcendental numbers. Ann. Inst. Fourier 56,
2093-2113 (2006).
[4] A. Baker, Continued fractions of transcendental numbers.
Mathematika 9, 1-8 (1962).
[5] V. Brun, La serie 1/5 + 1/7 +… est convergente ou finie. Bull.
Sci.Math. 43, 124-128 (1919).
[6] Y. Bugeaud, P. Hubert, and T. A. Schmidt, Transcen-
dence with Rosen continued fractions. ArXiv e-prints,
math.NT/1007.2050, Jul 2010.
[7] H. Davenport and K. F. Roth, Rational approximations to
algebraic numbers. Mathematika 2, 160-167 (1955).
[8] S. Finch, Mathematical Constants. Cambridge University
Press 2003.
[9] G. H. Hardy and E. M. Wright, An Introduction to the Theory
of Numbers. Oxford Science Publications 1980.
[10] J. Havil, Gamma: Exploring Euler’s Constant. Princeton
University Press, Princeton 2003.
[11] J. Kaczorowski, The boundary values of generalized Dirichlet
series and a problem of Chebyshev. Asterisque 209, 227-235
(1992).
[12] A. Y. Khinchin, Zur metrischen Kettenbruchtheorie. Compo-
sitio Mathematica 3, 275-286 (1936).
[13] A. Y. Khinchin, Continued Fractions. Dover Publications,
New York 1997.
[14] P. Lévy, Sur le développement en fraction continue d’un nom-
bre choisi au hasard. Compositio Mathematica 3, 286-303
(1936).
[15] E. Maillet, Introduction á la théorie des nombres transcen-
dants et des propriétés arithmétiques des fonctions. Gauthier-
Villars, Paris 1906.
[16] T. Nicely, Enumeration to 1.6 × 1015 of the twin primes and
Brun’s constant. http://www.trnicely.net/twins/twins2.html.
[17] M. Queffélec, Transcendance des fractions continues de Thue-
Morse. Journal of Number Theor 73, 201-211 (1998).
[18] P. Ribenboim, The Little Book of Big Primes. 2ed., Springer
2004.
[19] M. Rubinstein and P. Sarnak, Chebyshevs bias. Experimental
Mathematics 3, 173-197 (1994).
[20] C. Ryll-Nardzewski, On the ergodic theorems II (Ergodic
theory of continued fractions). Studia Mathematica 12, 74-79
(1951).
[21] M. R. Schroeder, Number Theory In Science And Communi-
cation, With Applications In Cryptography, Physics, Digital
Information, Computing, And Self-Similarity. Springer-Verlag
New York 2006.
[22] P. Sebah, Nmbrthry@listserv.nodak.edu mailing list, post da-
ted 22 Aug 2002. see also http://numbers.computation.free.fr/
Constants/Primes/twin.pdf.
[23] J. Sondow, Irrationality Measures, Irrationality Bases, and
a Theorem of Jarnik. http://arxiv.org/abs/math.NT/0406300
2004.
[24] The PARI Group, Bordeaux, PARI/GP, version 2.3.2, 2008.
available from pari.math.u-bordeaux.fr/.
[25] A. van der Poorten and J. Shallit, A specialised continued
fraction. Canad. J. Math. 45(5), 1067-1079 (1993).
[26] S. S. Wagstaff Jr, Divisors of Mersenne numbers. Mathematics
of Computation 40(161), 385-397 (1983).
[27] M. Wolf, Remark on the irrationality of the Brun’s constant.
ArXiv: math.NT/1002.4174 2010.
We have calculated on the computer the sum BM of reciprocals of first 47 known Mersenne primes with the accuracy of over 12000000 decimal digits. Next we developed BM into the continued fraction and calculated geometrical
means of the partial denominators of the continued fraction expansion of BM . We get values converging to the Khinchin’s constant. Next we calculated the n-th square roots of the denominators of the n-th convergents of these continued fractions obtaining values approaching the Khinchin-Lèvy constant. These two results suggests that the sum of reciprocals of all Mersenne primes is irrational, supporting the common belief that there is an infinity of the Mersenne primes. For comparison we have done the same procedures with a slightly modified set of 47 numbers obtaining quite different results. Next we investigated the continued fraction whose partial quotients are Mersenne primes and we argue that it should be transcendental.
Key words:
References:
[1] B. Adamczewski and Y. Bugeaud, On the Maillet-Baker
continued fractions. Journal für die reine und angewandte
Mathematik, 606, 105-121 (2007).
[2] B. Adamczewski and Y. Bugeaud, A short proof of the tran-
scendence of thue-morse continued fractions. American Math-
ematical Monthly 114, 536-540 (2007).
[3] B. Adamczewski, Y. Bugeaud, and L. Davison, Continued
fractions and transcendental numbers. Ann. Inst. Fourier 56,
2093-2113 (2006).
[4] A. Baker, Continued fractions of transcendental numbers.
Mathematika 9, 1-8 (1962).
[5] V. Brun, La serie 1/5 + 1/7 +… est convergente ou finie. Bull.
Sci.Math. 43, 124-128 (1919).
[6] Y. Bugeaud, P. Hubert, and T. A. Schmidt, Transcen-
dence with Rosen continued fractions. ArXiv e-prints,
math.NT/1007.2050, Jul 2010.
[7] H. Davenport and K. F. Roth, Rational approximations to
algebraic numbers. Mathematika 2, 160-167 (1955).
[8] S. Finch, Mathematical Constants. Cambridge University
Press 2003.
[9] G. H. Hardy and E. M. Wright, An Introduction to the Theory
of Numbers. Oxford Science Publications 1980.
[10] J. Havil, Gamma: Exploring Euler’s Constant. Princeton
University Press, Princeton 2003.
[11] J. Kaczorowski, The boundary values of generalized Dirichlet
series and a problem of Chebyshev. Asterisque 209, 227-235
(1992).
[12] A. Y. Khinchin, Zur metrischen Kettenbruchtheorie. Compo-
sitio Mathematica 3, 275-286 (1936).
[13] A. Y. Khinchin, Continued Fractions. Dover Publications,
New York 1997.
[14] P. Lévy, Sur le développement en fraction continue d’un nom-
bre choisi au hasard. Compositio Mathematica 3, 286-303
(1936).
[15] E. Maillet, Introduction á la théorie des nombres transcen-
dants et des propriétés arithmétiques des fonctions. Gauthier-
Villars, Paris 1906.
[16] T. Nicely, Enumeration to 1.6 × 1015 of the twin primes and
Brun’s constant. http://www.trnicely.net/twins/twins2.html.
[17] M. Queffélec, Transcendance des fractions continues de Thue-
Morse. Journal of Number Theor 73, 201-211 (1998).
[18] P. Ribenboim, The Little Book of Big Primes. 2ed., Springer
2004.
[19] M. Rubinstein and P. Sarnak, Chebyshevs bias. Experimental
Mathematics 3, 173-197 (1994).
[20] C. Ryll-Nardzewski, On the ergodic theorems II (Ergodic
theory of continued fractions). Studia Mathematica 12, 74-79
(1951).
[21] M. R. Schroeder, Number Theory In Science And Communi-
cation, With Applications In Cryptography, Physics, Digital
Information, Computing, And Self-Similarity. Springer-Verlag
New York 2006.
[22] P. Sebah, Nmbrthry@listserv.nodak.edu mailing list, post da-
ted 22 Aug 2002. see also http://numbers.computation.free.fr/
Constants/Primes/twin.pdf.
[23] J. Sondow, Irrationality Measures, Irrationality Bases, and
a Theorem of Jarnik. http://arxiv.org/abs/math.NT/0406300
2004.
[24] The PARI Group, Bordeaux, PARI/GP, version 2.3.2, 2008.
available from pari.math.u-bordeaux.fr/.
[25] A. van der Poorten and J. Shallit, A specialised continued
fraction. Canad. J. Math. 45(5), 1067-1079 (1993).
[26] S. S. Wagstaff Jr, Divisors of Mersenne numbers. Mathematics
of Computation 40(161), 385-397 (1983).
[27] M. Wolf, Remark on the irrationality of the Brun’s constant.
ArXiv: math.NT/1002.4174 2010.
