<abstract xmlns="http://www.w3.org/1999/xhtml">

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 <italic>k</italic> ∈ ℕ. Then <italic>k</italic> is prime if and only if there exists <italic>f, i, j, m, u</italic> ∈ ℕ+, <italic>r, s, t</italic> ∈ ℕ unknowns such that 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forma-2022-0020_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>D</mi><mi>F</mi><mi>I</mi><mi> </mi><mtext>is</mtext><mi> </mi><mtext>square</mtext><mi> </mi><mi> </mi><mi> </mi><mo>∧</mo><mi> </mi><mrow><mo>(</mo><mrow><msup><mrow><mi>M</mi></mrow><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><msup><mrow><mi>S</mi></mrow><mn>2</mn></msup><mo>+</mo><mn>1</mn><mi> </mi><mi> </mi><mtext>is</mtext><mi> </mi><mi> </mi><mtext>square</mtext><mi> </mi><mi> </mi><mo>∧</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>M</mi><mi>U</mi></mrow><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><msup><mrow><mi>T</mi></mrow><mn>2</mn></msup><mo>+</mo><mn>1</mn><mi> </mi><mi> </mi><mtext>is</mtext><mi> </mi><mi> </mi><mtext>square</mtext><mo>∧</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>4</mn><msup><mrow><mi>f</mi></mrow><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>-</mo><mi>m</mi><mi>S</mi><mi>T</mi><mi>U</mi></mrow><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mrow><mi>u</mi></mrow><mn>2</mn></msup><msup><mrow><mi>S</mi></mrow><mn>2</mn></msup><msup><mrow><mi>T</mi></mrow><mn>2</mn></msup><mo>&lt;</mo><mn>8</mn><mi>f</mi><mi>u</mi><mi>S</mi><mi>T</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>-</mo><mi>m</mi><mi>S</mi><mi>T</mi><mi>U</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>F</mi><mi>L</mi><mo>|</mo><mrow><mo>(</mo><mrow><mi>H</mi><mo>-</mo><mi>C</mi></mrow><mo>)</mo></mrow><mi>Z</mi><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mrow><mi>f</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>Q</mi><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><msup><mrow><mi>W</mi></mrow><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>S</mi><mi>u</mi><mo>-</mo><msup><mrow><mi>W</mi></mrow><mn>2</mn></msup><msup><mrow><mi>u</mi></mrow><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\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} &lt; 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 }</tex-math>
</alternatives>
</disp-formula>
where auxiliary variables <italic>A − I, L, M, S − W, Q</italic> ∈ ℤ are simply abbreviations defined as follows <italic>W</italic> = 100<italic>fk</italic>(<italic>k</italic> + 1), <italic>U</italic> = 100<italic>u</italic>3<italic>W</italic> 3 + 1, <italic>M</italic> = 100<italic>mUW</italic> + 1, <italic>S</italic> = (<italic>M −</italic>1)<italic>s</italic>+<italic>k</italic>+1, <italic>T</italic> = (<italic>MU −</italic>1)<italic>t</italic>+<italic>W −k</italic>+1, <italic>Q</italic> = 2<italic>MW −W</italic> 2<italic>−</italic>1, <italic>L</italic> = (<italic>k</italic>+1)<italic>Q</italic>, <italic>A</italic> = <italic>M</italic>(<italic>U</italic> +1), <italic>B</italic> = <italic>W</italic> +1, <italic>C</italic> = <italic>r</italic> +<italic>W</italic> +1, <italic>D</italic> = (<italic>A</italic>2 <italic>−</italic>1)<italic>C</italic>2 +1, <italic>E</italic> = 2<italic>iC</italic>2<italic>LD</italic>, <italic>F</italic> = (<italic>A</italic>2 <italic>−</italic>1)<italic>E</italic>2 +1, <italic>G</italic> = <italic>A</italic>+<italic>F</italic> (<italic>F −A</italic>), <italic>H</italic> = <italic>B</italic>+2(<italic>j −</italic>1)<italic>C</italic>, <italic>I</italic> = (<italic>G</italic>2 <italic>−</italic>1)<italic>H</italic>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 <italic>k</italic> this gives a total of 14 variables. This work has been partially presented in [8].
</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 



DFI is square   ∧ (M2-1)S2+1  is  square  ∧((MU)2-1)T2+1  is  square∧(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 2−1, 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 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].

Prime Representing Polynomial with 10 Unknowns – Introduction. Part II

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Prime Representing Polynomial with 10 Unknowns \u2013 Introduction. Part II"}

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...

Prime Representing Polynomial with 10 Unknowns – Introduction. Part II

Published Online: Feb 18, 2023

Page range: 245 - 253

Accepted: Dec 27, 2022

DOI: https://doi.org/10.2478/forma-2022-0020

Keywords
polynomial reduction, diophantine equation

© 2022 Karol Pąk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Prime Representing Polynomial with 10 Unknowns – Introduction. Part II

Published Online: Feb 18, 2023

Page range: 245 - 253

Accepted: Dec 27, 2022

DOI: https://doi.org/10.2478/forma-2022-0020

Keywordspolynomial reduction, diophantine equation

© 2022 Karol Pąk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
polynomial reduction, diophantine equation