ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 13 (2018), 171 -- 181
This work is licensed under a Creative Commons Attribution 4.0 International License.
ON THE NUMBERS THAT DETERMINE THE DISTRIBUTION OF TWIN PRIMES
Antonie Dinculescu
Abstract. This paper is about a class of numbers indirectly connected to the twin primes, which have not been investigated so far. With the help of these numbers, we look at the set of twin primes from a different perspective and bring the reader's attention to some characteristics potentially useful in tackling the Twin Prime Conjecture. To this purpose, we simplify the problem by dividing the positive integers into two complementary sets: one whose members lead to a pair of twin primes by a simple algebraic operation but cannot be directly calculated, and another one whose members do not lead directly to twin primes but are responsible for their distribution and can be directly calculated. By analyzing the facts from the perspective of the numbers in the second set, we reveal some interesting patterns and properties that suggest new approaches to the problem.
2010 Mathematics Subject Classification: 11A41.
Keywords: prime numbers; Twin Prime Conjecture; distribution of prime numbers; sieves
References
S. R. Finch, Hardy-Littlewood constants, \S 2.1 in Mathematical Constants. Encyclopedia of Mathematics and Its Applications 94, Cambridge University Press, Cambridge, 2003, 84-94. MR2003519. Zbl 1054.00001.
R. K. Guy, Unsolved Problems in Number Theory. 3rd ed. Problem Books in Mathematics, Springer-Verlag, New York, 2004, 32-35. MR2076335. Zbl 1058.11001.
G. H. Hardy and J. E. Littlewood, On some problems of `Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Matematica, 44 (1923), 1-70. MR1555183. JFM 48.0143.04.
J. Havil, Gamma. Exploring Euler's constant. With a foreword by Freeman Dyson, Princeton University Press, Princeton, 2003. MR1968276. Zbl 1375.26002.
F. Mertens, Ein Beitrag zur analytischen Zahlentheorie (A contribution to analytic number theory), Journal fur die Reine und Angewandte Mathematik 78 (1874), 46-62. MR1579612. JFM 06.0116.01.
H. Tietze, Famous Problems of Mathematics: Solved and unsolved mathematics problems from antiquity to modern times, Authorized translation from the second (1959) revised German edition. Edited by Beatrice Kevitt Hofstadter and Horace Komm. (English), Graylock Press, New York, 1965, 7-9. MR0181558. Zbl 0133.24102.
Antonie Dinculescu
4148, NW 34th Drive, Gainesville, Florida, 32605, USA.
email: adinculescu@cox.net