Volume 30 (2022): Issue 4 (December 2022)

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

4 Articles
Open Access

Prime Representing Polynomial with 10 Unknowns – Introduction. Part II

Published Online: 18 Feb 2023
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 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 DFIissquare(M2-1)S2+1issquare((MU)2-1)T2+1issquare(4f2-1)(r-mSTU)2+4u2S2T2<8fuST(r-mSTU)FL|(H-C)Z+F(f+1)Q+F(k+1)((W2-1)Su-W2u2+1) \matrix{ {DFI\,is\,square\,\,\,{\Lambda}\,\left( {{M^2} - 1} \right){S^2} + 1\,\,is\,\,square\,\,{\Lambda}} \hfill \cr {\left( {{{\left( {MU} \right)}^2} - 1} \right){T^2} + 1\,\,is\,\,square{\Lambda}} \hfill \cr {\left( {4{f^2} - 1} \right){{\left( {r - mSTU} \right)}^2} + 4{u^2}{S^2}{T^2} < 8fuST\left( {r - mSTU} \right)} \hfill \cr {FL|\left( {H - C} \right)Z + F\left( {f + 1} \right)Q + F\left( {k + 1} \right)\left( {\left( {{W^2} - 1} \right)Su - {W^2}{u^2} + 1} \right)} \hfill \cr } where auxiliary variables A − I, L, M, S − W, Q ∈ ℤ are simply abbreviations defined as follows W = 100fk(k + 1), U = 100u3W 3 + 1, M = 100mUW + 1, S = (M −1)s+k+1, T = (MU −1)t+W −k+1, Q = 2MW −W 21, 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 for each 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].

Keywords

• polynomial reduction
• diophantine equation

• 11D45
• 68V20
Open Access

Prime Representing Polynomial with 10 Unknowns

Published Online: 18 Feb 2023
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 i-th variable have the same number of variables. Then we study the number of variables that are used in given multivariate polynomials. By the used variable we mean a variable that is raised at least once to a non-zero power. We consider both adding unused variables and eliminating them.

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

• 11D45
• 68V20
Open Access

Existence and Uniqueness of Algebraic Closures

Published Online: 18 Feb 2023
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 F there exists a field extension E such that every non-constant polynomial pF [X] has a root in E. Artin’s proof applies Kronecker’s construction to each polynomial pF [X]\F simultaneously. To do so we needed the polynomial ring F [X1, X2, ...] with infinitely many variables, one for each polynomal pF [X]\F. The desired field extension E then is F [X1, X2, …]\I, where I is a maximal ideal generated by all non-constant polynomials pF [X]. Note, that to show that I is maximal Zorn’s lemma has to be applied.

In this second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial pA[X] has a root. The field of algebraic elements of A then is an algebraic closure of F. To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism FA, where A is an algebraic closure of F can be extended to a monomorphism EA, where E is any algebraic extension of F. In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn’s lemma.

Keywords

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

• 12F05
• 68V20
Open Access

Formalization of Orthogonal Decomposition for Hilbert Spaces

Published Online: 18 Feb 2023
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 S of a Hilbert space H, any vector can be represented by the sum of a vector in S and a vector orthogonal to S. The formalization of orthogonal complements of Hilbert spaces has been stored in the Mizar Mathematical Library [4]. We referred to [5] and [6] in the formalization.

Keywords

• Hilbert space
• orthogonal decomposition
• topological space

• 46Bxx
• 68V20