Article
Version 1
This version is not peer-reviewed
On the Natural Numbers that cannot be Expressed as a Sum of Two Primes
Version 1
: Received: 19 August 2024 / Approved: 20 August 2024 / Online: 20 August 2024 (12:03:53 CEST)
How to cite: Szabó, P. On the Natural Numbers that cannot be Expressed as a Sum of Two Primes. Preprints 2024, 2024081416. https://doi.org/10.20944/preprints202408.1416.v1 Szabó, P. On the Natural Numbers that cannot be Expressed as a Sum of Two Primes. Preprints 2024, 2024081416. https://doi.org/10.20944/preprints202408.1416.v1
Abstract
In this paper, a structural property of prime numbers will be introduced and analyzed. Our analysis is based on the application of the sequence of even integers 2,10,16,22,26,28… which is defined as follows:
F(0) = 2, for i > 0, F(i) = F(i − 1)+ l, where l is the smallest even integer that is valid for:
F(i − 1)+ 2k − 1 ∈ P, for k = 1, 2, . . . l/2-1,
F(i − 1)+ 2k − 1 ∉ P, for k = 2l
and P is the set of prime numbers.
The sequence F answers the question, which are the natural numbers that cannot be expressed as a
sum of two primes. We prove that numbers expressed as j = F(i)+ 1 cannot be written as the sum of
two primes, for i = 1, 2, ··· .
Keywords
circulant matrix; prime number; the Goldbach conjecture; the ternary Goldbach conjecture
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment
