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The Complexity of Mathematics
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: Received: 24 February 2020 / Approved: 25 February 2020 / Online: 25 February 2020 (12:21:49 CET)
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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 Mathematics. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v4 Vega, F. The Complexity of Mathematics. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v4
Abstract
The strong Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two primes. 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 principal 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 this happens, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. In addition, if this happens, then the Twin prime conjecture is true. Moreover, if this happens, then the Beal's conjecture is true. Furthermore, if this happens, then the Riemann hypothesis is true. Since the weak Goldbach's conjecture was proven, then this will certainly happen.
Keywords
complexity classes; regular languages; reduction; 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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