The Complexity of Number Theory

Frank Vega

doi:10.20944/preprints202002.0379.v1

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preprints.org > computer science and mathematics > computational mathematics > doi: 10.20944/preprints202002.0379.v1

Preprint Short Note Version 1 Preserved in Portico This version is not peer-reviewed

The Complexity of Number Theory

Frank Vega

Version 1 : Received: 24 February 2020 / Approved: 25 February 2020 / Online: 25 February 2020 (12:21:49 CET)
Version 2 : Received: 27 February 2020 / Approved: 27 February 2020 / Online: 27 February 2020 (10:49:49 CET)
Version 3 : Received: 10 March 2020 / Approved: 11 March 2020 / Online: 11 March 2020 (16:04:28 CET)
Version 4 : Received: 31 March 2020 / Approved: 2 April 2020 / Online: 2 April 2020 (18:25:32 CEST)
Version 5 : Received: 20 April 2020 / Approved: 22 April 2020 / Online: 22 April 2020 (09:48:30 CEST)
Version 6 : Received: 3 June 2020 / Approved: 4 June 2020 / Online: 4 June 2020 (13:22:40 CEST)
Version 7 : Received: 6 June 2020 / Approved: 8 June 2020 / Online: 8 June 2020 (10:31:19 CEST)
Version 8 : Received: 2 July 2021 / Approved: 6 July 2021 / Online: 6 July 2021 (12:38:05 CEST)
Version 9 : Received: 14 October 2021 / Approved: 14 October 2021 / Online: 14 October 2021 (14:15:38 CEST)

How to cite: Vega, F. The Complexity of Number Theory. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v1 Vega, F. The Complexity of Number Theory. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v1

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MDPI and ACS Style

Vega, F. The Complexity of Number Theory. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v1

APA Style

Vega, F. (2020). The Complexity of Number Theory. Preprints. https://doi.org/10.20944/preprints202002.0379.v1

Chicago/Turabian Style

Vega, F. 2020 "The Complexity of Number Theory" Preprints. https://doi.org/10.20944/preprints202002.0379.v1

Abstract

The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A major complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal's conjecture is true. Since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem.

Keywords

complexity classes; regular languages; number theory; conjecture; primes

Subject

Computer Science and Mathematics, Computational Mathematics

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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