Rule of Thumb Bounds in Goldbach’s Conjecture

Christopher Provatidis^1, , Emmanuel Markakis² and 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

Cite this paper:
Christopher Provatidis, Emmanuel Markakis and Nikiforos Markakis. Rule of Thumb Bounds in Goldbach’s Conjecture. American Journal of Mathematical Analysis. 2013; 1(1):8-13. doi: 10.12691/ajma-1-1-2

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

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Figures

References:

[1]	Goldbach, C., Letter to L. Euler, June 7, 1742.
[2]	Markakis, E., Provatidis, C. and Markakis, N., “An exploration on Goldbach’s conjecture,” International Journal of Pure and Applied Mathematics, 84(1), 2013.
[3]	Sylvester, J. J., “On the partition of an even number into two primes,” Proc. London Math. Soc., s1-4(1). 4-6. 1871.
[4]	G.H. Hardy, J.E. Littlewood, “Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes,” Acta Math., 44 (1). 1-70. 1923.
[5]	http://mathworld.wolfram.com/GoldbachConjecture.html
[6]	http://mathworld.wolfram.com/TwinPrimesConstant.html
[7]	Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974.
[8]	Cramér, H., “On the distribution of primes,” Proc. Camb. Phil. Soc., 20. 272-280. 1920.
[9]	Granville, A., “Harald Cramér and the Distribution of Prime Numbers,” Scand. Actuarial J., 1. 12-28. 1995.
[10]	http://mathworld.wolfram.com/k-TupleConjecture.html
[11]	P.T. Bateman, R. Horn, "A heuristic formula concerning the distribution of prime numbers" Math. Comp., 16. 363-367. 1962.
[12]	Friedlander J. and Iwaniec H., Opera de Cribro, Colloquium Publications, Volume 57, American Mathematical Society, 2010.
[13]	Cojocaru, A.C. and Murty, M.R., An introduction to sieve methods and their applications, Cambridge University Press, Cambridge, 2005.
[14]	Koukoulopoulos, D., MAT 6684W: Sieve methods, Université de Montréal, Fall 2012.
[15]	Helfgott, H.A., “Minor arcs for Goldbach’s problem,” May 2012.
[16]	Markakis, E., Provatidis, C. and Markakis, N., “Some issues on Goldbach Conjecture,” 29 May 2012.
[17]	Provatidis, C.G., “More properties in Goldbach’s Conjecture,” 22 October 2012.
[18]	Provatidis, C.G., “Smoothening Properties Related to the Goldbach’s Conjecture,” 13 October 2012.
[19]	Provatidis, C.G., “On the minimum number of pairs that fulfill Goldbach’s Conjecture,” 13 October 2012.