Formalization of the MRDP Theorem in the Mizar System

This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem.In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form x2 − (a2 − 1)y2 = 1 [8] and its solutions considered as two sequences {xi(a)}i=0∞,{yi(a)}i=0∞. We showed in [1] that the n-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation x = yi(a) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function y = xz is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9].In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that every recursively enumerable set is Diophantine, where recursively enumerable sets have been defined by the Martin Davis normal form.The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].Institute of Informatics, University of Białystok, PolandMarcin Acewicz and Karol Pąk. Basic Diophantine relations. Formalized Mathematics, 26(2):175–181, 2018. doi:10.2478/forma-2018-0015.Marcin Acewicz and Karol Pąk. Pell’s equation. Formalized Mathematics, 25(3):197–204, 2017. doi:10.1515/forma-2017-0019.Zofia Adamowicz and Paweł Zbierski. Logic of Mathematics: A Modern Course of Classical Logic. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley-Interscience, 1997.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.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.Martin Davis. Hilbert’s tenth problem is unsolvable. The American Mathematical Monthly, Mathematical Association of America, 80(3):233–269, 1973. doi:10.2307/2318447.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Karol Pąk. The Matiyasevich theorem. Preliminaries. Formalized Mathematics, 25(4): 315–322, 2017. doi:10.1515/forma-2017-0029.Karol Pąk. Diophantine sets. Part II. Formalized Mathematics, 27(2):197–208, 2019. doi:10.2478/forma-2019-0019.Craig Alan Smorynski. Logical Number Theory I, An Introduction. Universitext. Springer-Verlag Berlin Heidelberg, 1991. ISBN 978-3-642-75462-3.27220922

Prime Representing Polynomial

The main purpose of formalization is to prove that the set of prime numbers is diophantine, i.e., is representable by a polynomial formula. We formalize this problem, using the Mizar system [1], [2], in two independent ways, proving the existence of a polynomial without formulating it explicitly as well as with its indication. First, we reuse nearly all the techniques invented to prove the MRDPtheorem [11]. Applying a trick with Mizar schemes that go beyond first-order logic we give a short sophisticated proof for the existence of such a polynomial but without formulating it explicitly. Then we formulate the polynomial proposed in [6] that has 26 variables in the Mizar language as follows (w·z+h+j−q)²+((g·k+g+k)·(h+j)+h−z)²+(2 · k³·(2·k+2)·(n + 1)²+1−f²)²+(p+q+z + 2·n−e)² + (e³·(e+ 2)·(a + 1)² + 1−o²)² + (x² −(a² −´1)·y² −1)² +(16 ·(a² − 1)· r²· y²· y² + 1 − u²)² + (((a + u²·(u² − a))² − 1)·(n + 4 · d · y)² +1 − (x + c · u)²)² +(m² − (a² −´1) · l² − 1)² + (k + i · (a − 1) − l)² + (n + l + v − y)² +(p + l · (a − n − 1) + b · (2 · a · (n + 1) − (n + 1)² − 1) − m)² +(q + y · (a − p − 1) + s · (2 · a · (p + 1) − (p + 1)² − 1) − x)² + (z + p · l · (a − p) +t · (2 · a · p − p² − 1) − p · m)² and we prove that that for any positive integer k so that k + 1 is prime it is necessary and sufficient that there exist other natural variables a-z for which the polynomial equals zero. 26 variables is not the best known result in relation to the set of prime numbers, since any diophantine equation over N can be reduced to one in 13 unknowns [8] or even less [5], [13]. The best currently known result for all prime numbers, where the polynomial is explicitly constructed is 10 [7] or even 7 in the case of Fermat as well as Mersenne prime number [4]. We are currently focusing our formalization efforts in this direction.Institute of Computer Science, University of Białystok, PolandGrzegorz 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:10.1007/978-3-319-20615-817.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.Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Gaussian integers. Formalized Mathematics, 21(2):115–125, 2013. doi:10.2478/forma-2013-0013.James P. Jones. Diophantine representation of Mersenne and Fermat primes. Acta Arithmetica, 35:209–221, 1979. doi:10.4064/AA-35-3-209-221.James P. Jones. Universal diophantine equation. Journal of Symbolic Logic, 47(4):549–571, 1982.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.Yuri Matiyasevich. Primes are nonnegative values of a polynomial in 10 variables. Journal of Soviet Mathematics, 15:33–44, 1981. doi:10.1007/BF01404106.Yuri Matiyasevich and Julia Robinson. Reduction of an arbitrary diophantine equation to one in 13 unknowns. Acta Arithmetica, 27:521–553, 1975.Karol Pąk. The Matiyasevich theorem. Preliminaries. Formalized Mathematics, 25(4):315–322, 2017. doi:10.1515/forma-2017-0029.Karol Pąk. Diophantine sets. Part II. Formalized Mathematics, 27(2):197–208, 2019. doi:10.2478/forma-2019-0019.Karol Pąk. Formalization of the MRDP theorem in the Mizar system. Formalized Mathematics, 27(2):209–221, 2019. doi:10.2478/forma-2019-0020.Christoph Schwarzweller. Proth numbers. Formalized Mathematics, 22(2):111–118, 2014. doi:10.2478/forma-2014-0013.Zhi-Wei Sun. Further results on Hilbert’s Tenth Problem. Science China Mathematics, 64:281–306, 2021. doi:10.1007/s11425-020-1813-5.Rafał Ziobro. Prime factorization of sums and differences of two like powers. Formalized Mathematics, 24(3):187–198, 2016. doi:10.1515/forma-2016-0015.29422122