eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-08-03
26:1
26:8
10.4230/LIPIcs.ITP.2022.26
article
Formalizing a Diophantine Representation of the Set of Prime Numbers
Pąk, Karol
1
https://orcid.org/0000-0002-7099-1669
Kaliszyk, Cezary
2
https://orcid.org/0000-0002-8273-6059
University of Białystok, Poland
Universität Innsbruck, Austria
The DPRM (Davis-Putnam-Robinson-Matiyasevich) theorem is the main step in the negative resolution of Hilbert’s 10th problem. Almost three decades of work on the problem have resulted in several equally surprising results. These include the existence of diophantine equations with a reduced number of variables, as well as the explicit construction of polynomials that represent specific sets, in particular the set of primes. In this work, we formalize these constructions in the Mizar system. We focus on the set of prime numbers and its explicit representation using 10 variables. It is the smallest representation known today. For this, we show that the exponential function is diophantine, together with the same properties for the binomial coefficient and factorial. This formalization is the next step in the research on formal approaches to diophantine sets following the DPRM theorem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol237-itp2022/LIPIcs.ITP.2022.26/LIPIcs.ITP.2022.26.pdf
DPRM theorem
Polynomial reduction
prime numbers