<abstract xmlns="http://www.w3.org/1999/xhtml">The article is the next in a series aiming to formalize the MDPR-theorem using the Mizar proof assistant [3], [6], [4]. We analyze four equations from the Diophantine standpoint that are crucial in the bounded quantifier theorem, that is used in one of the approaches to solve the problem.Based on our previous work [1], we prove that the value of a given binomial coefficient and factorial can be determined by its arguments in a Diophantine way. Then we prove that two products<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forma-2019-0019_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>z</mi><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>x</mi></munderover><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>i</mi><mo>⋅</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi></mrow><mi>z</mi><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>x</mi></munderover><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn><mo>-</mo><mi>j</mi></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mo>(</mo><mn>0.1</mn><mo>)</mo></mrow></mrow></math><tex-math>$$z = \prod\limits_{i = 1}^x {\left( {1 + i \cdot y} \right),\,\,\,\,\,\,\,\,} z = \prod\limits_{i = 1}^x {\left( {y + 1 - j} \right),\,\,\,\,\,\,(0.1)} $$</tex-math></alternatives></disp-formula>where <italic>y > x</italic> are Diophantine.The formalization follows [10], Z. Adamowicz, P. Zbierski [2] as well as M. Davis [5].</abstract>

The article is the next in a series aiming to formalize the MDPR-theorem using the Mizar proof assistant [3], [6], [4]. We analyze four equations from the Diophantine standpoint that are crucial in the bounded quantifier theorem, that is used in one of the approaches to solve the problem.Based on our previous work [1], we prove that the value of a given binomial coefficient and factorial can be determined by its arguments in a Diophantine way. Then we prove that two productsz=∏i=1x(1+i⋅y),        z=∏i=1x(y+1-j),      (0.1)$$z = \prod\limits_{i = 1}^x {\left( {1 + i \cdot y} \right),\,\,\,\,\,\,\,\,} z = \prod\limits_{i = 1}^x {\left( {y + 1 - j} \right),\,\,\,\,\,\,(0.1)} $$where y > x are Diophantine.The formalization follows [10], Z. Adamowicz, P. Zbierski [2] as well as M. Davis [5].

Diophantine Sets. Part II

Formalized Mathematics

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{"article-title":"Diophantine Sets. Part II"}

The article is the next in a series aiming to formalize the MDPR-theorem using the Mizar proof assistant [3], [6], [4]. We analyze four equations from the...

丢番图集。第二部分

在线发布:2019年7月20日

页面范围： 197 - 208

认可的:2019年5月27日

内政部： https://doi.org/10.2478/forma-2019-0019

关键词
希尔伯特的第十个问题,丢番图关系

©2019 Karol Pąk，由Sciendo出版

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