Open Access Peer-reviewed

Rule of Thumb Bounds in Goldbach’s Conjecture

Christopher Provatidis1,, Emmanuel Markakis2, Nikiforos Markakis3

1Department of Mechanical Engineering, National Technical University of Athens, Athens, Greece

2Vassilissis Olgas 129B, 54643 Thessaloniki, Greece

3Cram school “Methodiko”, Vouliagmenis and Kyprou 2, 16452 Argyroupolis, Greece

American Journal of Mathematical Analysis. 2013, 1(1), 8-13. DOI: 10.12691/ajma-1-1-2
Published online: August 25, 2017

Abstract

This paper determines proper factors for old and novel logarithmic functions previously used in asymptotic formulas, to make them conservative lower bounds for the “thumb-of-rule” estimation of the number of representations of an even number 2n as a sum of two odd primes (Goldbach’s conjecture). Numerical experiments up to 2n = 500,000 show that, in the graph of the number of prime-pairs versus 2n, the ratio of the ordinate of lowest “cloud” points over the aforementioned functions tends asymptotically to values between 0.61 and 0.74. One of the three formulas proposed takes the simple form 4n/[3(lnn)2], which is a conservative lower bound for the number of representations of an even number 2n as a sum of two odd primes.

Keywords:

Goldbach’s conjecture, probabilistic number theory, lower bound
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