### Rule of Thumb Bounds in Goldbach’s Conjecture

**Christopher Provatidis**^{1,}, **Emmanuel Markakis**^{2}, **Nikiforos Markakis**^{3}

^{1}Department of Mechanical Engineering, National Technical University of Athens, Athens, Greece

^{2}Vassilissis Olgas 129B, 54643 Thessaloniki, Greece

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

*American Journal of Mathematical Analysis*, **2013** 1 (1),
pp 8-13

DOI: 10.12691/ajma-1-1-2

Received January 03, 2013; Revised February 23, 2013; Accepted February 25, 2013

Corresponding author: cprovat@central.ntua.gr |

## Cite This Article:

- Provatidis, Christopher, Emmanuel Markakis, and Nikiforos Markakis. "Rule of Thumb Bounds in Goldbach’s Conjecture."
*American Journal of Mathematical Analysis*1.1 (2013): 8-13.

- Provatidis, C. , Markakis, E. , & Markakis, N. (2013). Rule of Thumb Bounds in Goldbach’s Conjecture.
*American Journal of Mathematical Analysis*,*1*(1), 8-13.

- Provatidis, Christopher, Emmanuel Markakis, and Nikiforos Markakis. "Rule of Thumb Bounds in Goldbach’s Conjecture."
*American Journal of Mathematical Analysis*1, no. 1 (2013): 8-13.

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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.

Goldbach’s conjecture, probabilistic number theory, lower bound

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