REPOZYTORIUM UNIWERSYTETU
W BIAŁYMSTOKU

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 Tytuł: Prime Representing Polynomial with 10 Unknowns – Introduction. Part II Autorzy: Pąk, Karol Słowa kluczowe: polynomial reductiondiophantine equation Data wydania: 2022 Data dodania: 27-lut-2023 Wydawca: DeGruyter Open Źródło: Formalized Mathematics, Volume 30, Issue 4, Pages 245-253 Abstrakt: 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]. Afiliacja: Institute of Computer Science, University of Białystok, Poland URI: http://hdl.handle.net/11320/14705 DOI: 10.2478/forma-2022-0020 ISSN: 1426-2630 e-ISSN: 1898-9934 Typ Dokumentu: Article metadata.dc.rights.uri: https://creativecommons.org/licenses/by-sa/3.0/ Właściciel praw: © 2022 The Author(s)CC BY-SA 3.0 license Występuje w kolekcji(ach): Artykuły naukowe (WInf)Formalized Mathematics, 2022, Volume 30, Issue 4

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