Preprint Article Version 5 Preserved in Portico This version is not peer-reviewed

The Complexity of Mathematics

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 Mathematics. Preprints 2020, 2020020379. Vega, F. The Complexity of Mathematics. Preprints 2020, 2020020379.


In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.


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


Computer Science and Mathematics, Computational Mathematics

Comments (1)

Comment 1
Received: 22 April 2020
Commenter: Frank Vega
Commenter's Conflict of Interests: Author
Comment: This new version is an extension of the older version. Certainly, in this version we found another way to prove the problem PRIMES is not in 1NSPACE(S(n)) for all S(n). In this way, the proof of Riemann hypothesis and Goldbach's conjecture are still valid according to the older version, because they are supported in that previous proof. Indeed, we found the other problems in the older version have not a strong proof. For that reason, they were removed from this new version. Moreover, we improve the proof of the Goldbach's conjecture. On the other hand, the proof of the Riemann hypothesis remains almost the same. The abstract and content of the paper were changed and improved.
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