Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>=</mo><mfenced separators="" open="(" close=")"><msub><mi>s</mi&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/18/4006 |
| _version_ | 1797578963154894848 |
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| author | Riccardo Gatti |
| author_facet | Riccardo Gatti |
| author_sort | Riccardo Gatti |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>=</mo><mfenced separators="" open="(" close=")"><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>n</mi></msub></mfenced></mrow></semantics></math></inline-formula> be a finite sequence of integers. Then, <i>S</i> is a Gilbreath sequence of length <i>n</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>∈</mo><msub><mi mathvariant="double-struck">G</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>, iff <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mn>1</mn></msub></semantics></math></inline-formula> is even or odd and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> are, respectively, odd or even and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>min</mi><msub><mi mathvariant="double-struck">K</mi><mfenced separators="" open="(" close=")"><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>m</mi></msub></mfenced></msub><mo>≤</mo><msub><mi>s</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≤</mo><mi>max</mi><msub><mi mathvariant="double-struck">K</mi><mfenced separators="" open="(" close=")"><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>m</mi></msub></mfenced></msub><mo>,</mo><mo>∀</mo><mi>m</mi><mo>∈</mo><mfenced separators="" open="[" close=")"><mn>1</mn><mo>,</mo><mi>n</mi></mfenced></mrow></semantics></math></inline-formula>. This, applied to the order sequence of prime number <i>P</i>, defines Gilbreath polynomials and two integer sequences, A347924 and A347925, which are used to prove that Gilbreath conjecture <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>C</mi></mrow></semantics></math></inline-formula> is implied by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>−</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⩽</mo><msub><mi mathvariant="script">P</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mfenced open="(" close=")"><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">P</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mfenced open="(" close=")"><mn>1</mn></mfenced></mrow></semantics></math></inline-formula> is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>-th Gilbreath polynomial at 1. |
| first_indexed | 2024-03-10T22:29:07Z |
| format | Article |
| id | doaj.art-7fc88c98ba244b6b852e13f77597eb58 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T22:29:07Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7fc88c98ba244b6b852e13f77597eb582023-11-19T11:50:37ZengMDPI AGMathematics2227-73902023-09-011118400610.3390/math11184006Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath ConjectureRiccardo Gatti0National Laboratory of Molecular Biology and Stem Cell Engineering, Istituto Nazionale di Biostrutture e Biosistemi (INBB) c/o Eldor Lab, Via di Corticella 183, 40128 Bologna, ItalyLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>=</mo><mfenced separators="" open="(" close=")"><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>n</mi></msub></mfenced></mrow></semantics></math></inline-formula> be a finite sequence of integers. Then, <i>S</i> is a Gilbreath sequence of length <i>n</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>∈</mo><msub><mi mathvariant="double-struck">G</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>, iff <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mn>1</mn></msub></semantics></math></inline-formula> is even or odd and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> are, respectively, odd or even and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>min</mi><msub><mi mathvariant="double-struck">K</mi><mfenced separators="" open="(" close=")"><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>m</mi></msub></mfenced></msub><mo>≤</mo><msub><mi>s</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≤</mo><mi>max</mi><msub><mi mathvariant="double-struck">K</mi><mfenced separators="" open="(" close=")"><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>s</mi><mi>m</mi></msub></mfenced></msub><mo>,</mo><mo>∀</mo><mi>m</mi><mo>∈</mo><mfenced separators="" open="[" close=")"><mn>1</mn><mo>,</mo><mi>n</mi></mfenced></mrow></semantics></math></inline-formula>. This, applied to the order sequence of prime number <i>P</i>, defines Gilbreath polynomials and two integer sequences, A347924 and A347925, which are used to prove that Gilbreath conjecture <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>C</mi></mrow></semantics></math></inline-formula> is implied by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>−</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⩽</mo><msub><mi mathvariant="script">P</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mfenced open="(" close=")"><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">P</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mfenced open="(" close=")"><mn>1</mn></mfenced></mrow></semantics></math></inline-formula> is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>-th Gilbreath polynomial at 1.https://www.mdpi.com/2227-7390/11/18/4006Gilbreath conjectureprime numberssequence of integer numbersGilbreath polynomials |
| spellingShingle | Riccardo Gatti Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture Mathematics Gilbreath conjecture prime numbers sequence of integer numbers Gilbreath polynomials |
| title | Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture |
| title_full | Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture |
| title_fullStr | Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture |
| title_full_unstemmed | Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture |
| title_short | Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture |
| title_sort | gilbreath equation gilbreath polynomials and upper and lower bounds for gilbreath conjecture |
| topic | Gilbreath conjecture prime numbers sequence of integer numbers Gilbreath polynomials |
| url | https://www.mdpi.com/2227-7390/11/18/4006 |
| work_keys_str_mv | AT riccardogatti gilbreathequationgilbreathpolynomialsandupperandlowerboundsforgilbreathconjecture |