Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes
The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the M...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/6/540 |
| _version_ | 1797596076296896512 |
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| author | Menachem Shlossberg |
| author_facet | Menachem Shlossberg |
| author_sort | Menachem Shlossberg |
| collection | DOAJ |
| description | The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">F</mi></semantics></math></inline-formula> is a subfield of a local field of characteristic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>≠</mo><mn>2</mn></mrow></semantics></math></inline-formula>, then the special upper triangular group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ST</mi><mo>+</mo></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">F</mi><mo>)</mo></mrow></semantics></math></inline-formula> is minimal precisely when the special linear group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">SL</mo><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">F</mi><mo>)</mo></mrow></semantics></math></inline-formula> is. We provide criteria for the minimality (and total minimality) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">SL</mo><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">F</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ST</mi><mo>+</mo></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">F</mi><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">F</mi></semantics></math></inline-formula> is a subfield of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">F</mi><mi>π</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">F</mi><mi>c</mi></msub></semantics></math></inline-formula> be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">A</mi><mo>∈</mo><mo>{</mo><msub><mi mathvariant="script">F</mi><mi>π</mi></msub><mo>,</mo><msub><mi mathvariant="script">F</mi><mi>c</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> is finite; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∏</mo><mrow><msub><mi mathvariant="double-struck">F</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="script">A</mi></mrow></msub><mo form="prefix">SL</mo><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>−</mo><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is minimal, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Q</mi><mo>(</mo><mi>i</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the Gaussian rational field; and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∏</mo><mrow><msub><mi mathvariant="double-struck">F</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="script">A</mi></mrow></msub><msup><mi>ST</mi><mo>+</mo></msup><mrow><mo>(</mo><msub><mi mathvariant="double-struck">F</mi><mi>n</mi></msub><mo>−</mo><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is minimal. Similarly, denote by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">M</mi><mi>π</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">M</mi><mi>c</mi></msub></semantics></math></inline-formula> the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>∈</mo><mo>{</mo><msub><mi mathvariant="script">M</mi><mi>π</mi></msub><mo>,</mo><msub><mi mathvariant="script">M</mi><mi>c</mi></msub><mo>}</mo><mo>.</mo></mrow></semantics></math></inline-formula> Then the following conditions are equivalent: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">B</mi></semantics></math></inline-formula> is finite; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∏</mo><mrow><msub><mi>M</mi><mi>p</mi></msub><mo>∈</mo><mi mathvariant="script">B</mi></mrow></msub><mo form="prefix">SL</mo><mrow><mo>(</mo><msub><mi>M</mi><mi>p</mi></msub><mo>+</mo><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is minimal; and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∏</mo><mrow><msub><mi>M</mi><mi>p</mi></msub><mo>∈</mo><mi mathvariant="script">B</mi></mrow></msub><msup><mi>ST</mi><mo>+</mo></msup><mrow><mo>(</mo><msub><mi>M</mi><mi>p</mi></msub><mo>+</mo><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is minimal. |
| first_indexed | 2024-03-11T02:46:28Z |
| format | Article |
| id | doaj.art-1a0b5195b3734d57bd85e76f83a9fc02 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T02:46:28Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-1a0b5195b3734d57bd85e76f83a9fc022023-11-18T09:16:30ZengMDPI AGAxioms2075-16802023-05-0112654010.3390/axioms12060540Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne PrimesMenachem Shlossberg0School of Computer Science, Reichman University, Herzliya 4610101, IsraelThe question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">F</mi></semantics></math></inline-formula> is a subfield of a local field of characteristic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>≠</mo><mn>2</mn></mrow></semantics></math></inline-formula>, then the special upper triangular group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ST</mi><mo>+</mo></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">F</mi><mo>)</mo></mrow></semantics></math></inline-formula> is minimal precisely when the special linear group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">SL</mo><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">F</mi><mo>)</mo></mrow></semantics></math></inline-formula> is. We provide criteria for the minimality (and total minimality) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">SL</mo><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">F</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ST</mi><mo>+</mo></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">F</mi><mo>)</mo><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">F</mi></semantics></math></inline-formula> is a subfield of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">F</mi><mi>π</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">F</mi><mi>c</mi></msub></semantics></math></inline-formula> be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">A</mi><mo>∈</mo><mo>{</mo><msub><mi mathvariant="script">F</mi><mi>π</mi></msub><mo>,</mo><msub><mi mathvariant="script">F</mi><mi>c</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> is finite; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∏</mo><mrow><msub><mi mathvariant="double-struck">F</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="script">A</mi></mrow></msub><mo form="prefix">SL</mo><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>−</mo><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is minimal, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Q</mi><mo>(</mo><mi>i</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the Gaussian rational field; and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∏</mo><mrow><msub><mi mathvariant="double-struck">F</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="script">A</mi></mrow></msub><msup><mi>ST</mi><mo>+</mo></msup><mrow><mo>(</mo><msub><mi mathvariant="double-struck">F</mi><mi>n</mi></msub><mo>−</mo><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is minimal. Similarly, denote by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">M</mi><mi>π</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">M</mi><mi>c</mi></msub></semantics></math></inline-formula> the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">B</mi><mo>∈</mo><mo>{</mo><msub><mi mathvariant="script">M</mi><mi>π</mi></msub><mo>,</mo><msub><mi mathvariant="script">M</mi><mi>c</mi></msub><mo>}</mo><mo>.</mo></mrow></semantics></math></inline-formula> Then the following conditions are equivalent: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">B</mi></semantics></math></inline-formula> is finite; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∏</mo><mrow><msub><mi>M</mi><mi>p</mi></msub><mo>∈</mo><mi mathvariant="script">B</mi></mrow></msub><mo form="prefix">SL</mo><mrow><mo>(</mo><msub><mi>M</mi><mi>p</mi></msub><mo>+</mo><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is minimal; and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∏</mo><mrow><msub><mi>M</mi><mi>p</mi></msub><mo>∈</mo><mi mathvariant="script">B</mi></mrow></msub><msup><mi>ST</mi><mo>+</mo></msup><mrow><mo>(</mo><msub><mi>M</mi><mi>p</mi></msub><mo>+</mo><mn>1</mn><mo>,</mo><mi mathvariant="double-struck">Q</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is minimal.https://www.mdpi.com/2075-1680/12/6/540Fermat primesFermat numbersMersenne primesminimal groupspecial linear groupGaussian rational field |
| spellingShingle | Menachem Shlossberg Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes Axioms Fermat primes Fermat numbers Mersenne primes minimal group special linear group Gaussian rational field |
| title | Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes |
| title_full | Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes |
| title_fullStr | Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes |
| title_full_unstemmed | Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes |
| title_short | Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes |
| title_sort | minimality conditions equivalent to the finitude of fermat and mersenne primes |
| topic | Fermat primes Fermat numbers Mersenne primes minimal group special linear group Gaussian rational field |
| url | https://www.mdpi.com/2075-1680/12/6/540 |
| work_keys_str_mv | AT menachemshlossberg minimalityconditionsequivalenttothefinitudeoffermatandmersenneprimes |