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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.
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