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The Complexity of Number Theory
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: 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)
Version 10 : Received: 20 June 2024 / Approved: 21 June 2024 / Online: 21 June 2024 (10:43:53 CEST)
Version 11 : Received: 23 June 2024 / Approved: 24 June 2024 / Online: 24 June 2024 (08:58:02 CEST)
Version 12 : Received: 25 June 2024 / Approved: 25 June 2024 / Online: 25 June 2024 (10:26:09 CEST)
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)
Version 10 : Received: 20 June 2024 / Approved: 21 June 2024 / Online: 21 June 2024 (10:43:53 CEST)
Version 11 : Received: 23 June 2024 / Approved: 24 June 2024 / Online: 24 June 2024 (08:58:02 CEST)
Version 12 : Received: 25 June 2024 / Approved: 25 June 2024 / Online: 25 June 2024 (10:26:09 CEST)
How to cite: Vega, F. The Complexity of Number Theory. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v2 Vega, F. The Complexity of Number Theory. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v2
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 classes are L, NL and NSPACE(S(n)) for some S(n). Whether L = NL is a fundamental question that it is as important as it is unresolved. 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). This proof is based on the assumption that if some language belongs to NSPACE(S(n)), then the unary version of that language belongs to NSPACE(S(log n)) and vice versa. 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 when L = NL. On November 2019, Frank Vega proves that L = NP which also implies that L = NL. In this way, the Beal's conjecture is true and 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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