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Pole DC | Wartość | Język |
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dc.contributor.author | Pąk, Karol | - |
dc.date.accessioned | 2023-02-27T11:06:49Z | - |
dc.date.available | 2023-02-27T11:06:49Z | - |
dc.date.issued | 2022 | - |
dc.identifier.citation | Formalized Mathematics, Volume 30, Issue 4, Pages 245-253 | pl |
dc.identifier.issn | 1426-2630 | - |
dc.identifier.uri | http://hdl.handle.net/11320/14705 | - |
dc.description.abstract | In our previous work [7] we prove that the set of prime numbersis diophantine using the 26-variable polynomial proposed in [4]. In this paper,we focus on the reduction of the number of variables to 10 and it is the smal-lest variables number known today [5], [10]. Using the Mizar [3], [2] system, weformalize the first step in this direction by proving Theorem 1 [5] formulated asfollows: Let k ∈ N. Then k is prime if and only if there exists f, i, j, m, u ∈ N+,r, s, t ∈ N unknowns such that DFI is square ∧(M²−1)S²+1 is square ∧((M U)²−1)T²+1 is square ∧(4f²−1)(r−mST U)²+ 4u²S²T²<8f uST(r−mST U)F L|(H−C)Z+F(f+ 1)Q+F(k+ 1)((W²−1)Su−W²u²+ 1) (0.1) where auxiliary variables A−I, L, M, S−W, Q ∈ Z are simply abbreviations defined as follows W= 100f k(k+ 1),U= 100u³W³+ 1,M= 100mU W+ 1,S= (M−1)s+k+1,T= (M U−1)t+W−k+1,Q= 2M W−W²−1,L= (k+1)Q,A=M(U+ 1),B=W+ 1,C=r+W+ 1,D= (A²−1)C²+ 1,E= 2iC²LD,F= (A²−1)E²+1,G=A+F(F−A),H=B+2(j−1)C,I= (G²−1)H²+1. It is easily see that (0.1) uses 8 unknowns explicitly along with five implicit one foreach diophantine relationship: is square, inequality, and divisibility. Together with k this gives a total of 14 variables. This work has been partially presented in [8]. | pl |
dc.language.iso | en | pl |
dc.publisher | DeGruyter Open | pl |
dc.rights | Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) | pl |
dc.rights.uri | https://creativecommons.org/licenses/by-sa/3.0/ | pl |
dc.subject | polynomial reduction | pl |
dc.subject | diophantine equation | pl |
dc.title | Prime Representing Polynomial with 10 Unknowns – Introduction. Part II | pl |
dc.type | Article | pl |
dc.rights.holder | © 2022 The Author(s) | pl |
dc.rights.holder | CC BY-SA 3.0 license | pl |
dc.identifier.doi | 10.2478/forma-2022-0020 | - |
dc.description.Affiliation | Institute of Computer Science, University of Białystok, Poland | pl |
dc.description.references | Marcin Acewicz and Karol Pąk. Pell’s equation. Formalized Mathematics, 25(3):197–204,2017. doi:10.1515/forma-2017-0019. | pl |
dc.description.references | Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors,Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17. | pl |
dc.description.references | Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32,2018. doi:10.1007/s10817-017-9440-6. | pl |
dc.description.references | James P. Jones, Sato Daihachiro, Hideo Wada, and Douglas Wiens. Diophantine representation of the set of prime numbers. The American Mathematical Monthly, 83(6):449–464,1976. | pl |
dc.description.references | Yuri Matiyasevich. Primes are nonnegative values of a polynomial in 10 variables. Journalof Soviet Mathematics, 15:33–44, 1981. doi:10.1007/BF01404106. | pl |
dc.description.references | Karol Pąk. The Matiyasevich theorem. Preliminaries. Formalized Mathematics, 25(4):315–322, 2017. doi:10.1515/forma-2017-0029. | pl |
dc.description.references | Karol Pąk. Prime representing polynomial. Formalized Mathematics, 29(4):221–228, 2021.doi:10.2478/forma-2021-0020. | pl |
dc.description.references | Karol Pąk and Cezary Kaliszyk. Formalizing a diophantine representation of the set of prime numbers. In June Andronick and Leonardo de Moura, editors,13th International Conference on Interactive Theorem Proving, ITP 2022, August 7-10, 2022, Haifa, Israel, volume 237 of LIPIcs, pages 26:1–26:8. Schloss Dagstuhl - Leibniz-Zentrum für Informatik,2022. doi:10.4230/LIPIcs.ITP.2022.26. | pl |
dc.description.references | Marco Riccardi. The perfect number theorem and Wilson’s theorem. Formalized Mathematics, 17(2):123–128, 2009. doi:10.2478/v10037-009-0013-y. | pl |
dc.description.references | Zhi-Wei Sun. Further results on Hilbert’s Tenth Problem. Science China Mathematics,64:281–306, 2021. doi:10.1007/s11425-020-1813-5. | pl |
dc.identifier.eissn | 1898-9934 | - |
dc.description.volume | 30 | pl |
dc.description.issue | 4 | pl |
dc.description.firstpage | 245 | pl |
dc.description.lastpage | 253 | pl |
dc.identifier.citation2 | Formalized Mathematics | pl |
Występuje w kolekcji(ach): | Artykuły naukowe (WInf) Formalized Mathematics, 2022, Volume 30, Issue 4 |
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