- Journal Details
- Format
- Journal
- eISSN
- 1898-9934
- First Published
- 09 Jun 2008
- Publication timeframe
- 4 times per year
- Languages
- English

#### Search

Open Access

#### Prime Representing Polynomial with 10 Unknowns – Introduction. Part II

Page range: 245 - 253

#### Abstract

In our previous work [7] we prove that the set of prime numbers is 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 smallest variables number known today [5], [10]. Using the Mizar [3], [2] system, we formalize the first step in this direction by proving Theorem 1 [5] formulated as follows: Let ^{+}, ^{3}^{3} + 1, ^{2}^{2} ^{2} +1, ^{2}^{2} ^{2} +1, ^{2} ^{2} +1. It is easily see that (0.1) uses 8 unknowns explicitly along with five implicit one for each diophantine relationship: is square, inequality, and divisibility. Together with

#### Keywords

- polynomial reduction
- diophantine equation

#### MSC

- 11D45
- 68V20

Open Access

#### Prime Representing Polynomial with 10 Unknowns

Page range: 255 - 279

#### Abstract

In this article we formalize in Mizar [1], [2] the final step of our attempt to formally construct a prime representing polynomial with 10 variables proposed by Yuri Matiyasevich in [4].

The first part of the article includes many auxiliary lemmas related to multivariate polynomials. We start from the properties of monomials, among them their evaluation as well as the power function on polynomials to define the substitution for multivariate polynomials. For simplicity, we assume that a polynomial and substituted ones as

The second part of the paper deals with the construction of the polynomial proposed by Yuri Matiyasevich. First, we introduce a diophantine polynomial over 4 variables that has roots in integers if and only if indicated variable is the square of a natural number, and another two is the square of an odd natural number. We modify the polynomial by adding two variables in such a way that the root additionally requires the divisibility of these added variables. Then we modify again the polynomial by adding two variables to also guarantee the nonnegativity condition of one of these variables. Finally, we combine the prime diophantine representation proved in [7] with the obtained polynomial constructing a prime representing polynomial with 10 variables. This work has been partially presented in [8] with the obtained polynomial constructing a prime representing polynomial with 10 variables in Theorem (85).

#### Keywords

- polynomial reduction
- prime representing polynomial

#### MSC

- 11D45
- 68V20

Open Access

#### Existence and Uniqueness of Algebraic Closures

Page range: 281 - 294

#### Abstract

This is the second part of a two-part article formalizing existence and uniqueness of algebraic closures, using the Mizar [2], [1] formalism. Our proof follows Artin’s classical one as presented by Lang in [3]. In the first part we proved that for a given field _{1}, _{2}, ...] with infinitely many variables, one for each polynomal _{1}, _{2}, …]

In this second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension

#### Keywords

- algebraic closures
- polynomial rings with countably infinite number of variables
- Emil Artin

#### MSC

- 12F05
- 68V20

Open Access

#### Formalization of Orthogonal Decomposition for Hilbert Spaces

Page range: 295 - 299

#### Abstract

In this article, we formalize the theorems about orthogonal decomposition of Hilbert spaces, using the Mizar system [1], [2]. For any subspace

#### Keywords

- Hilbert space
- orthogonal decomposition
- topological space

#### MSC

- 46Bxx
- 68V20