Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers
The classical problem of finding all integers <i>a</i> and <i>M</i> such that the sums of <i>M</i> consecutive squared integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><m...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/2/146 |
| _version_ | 1797296962302640128 |
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| author | Vladimir Pletser |
| author_facet | Vladimir Pletser |
| author_sort | Vladimir Pletser |
| collection | DOAJ |
| description | The classical problem of finding all integers <i>a</i> and <i>M</i> such that the sums of <i>M</i> consecutive squared integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mfenced separators="" open="(" close=")"><mi>a</mi><mo>+</mo><mi>i</mi></mfenced><mn>2</mn></msup></semantics></math></inline-formula> equal the squared integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>s</mi><mn>2</mn></msup></semantics></math></inline-formula>, where <i>M</i> is the number of terms in the sum, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>a</mi><mn>2</mn></msup></semantics></math></inline-formula> the first term and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>M</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, yields remarkable regular linear features when plotting the values of <i>M</i> as a function of <i>a</i>. These linear features correspond to groupings of pairs of <i>a</i> values for successive same values of <i>M</i> found on either side of straight lines of equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mi>M</mi><mo>=</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula>, where <i>c</i> is an integer constant and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> a parameter taking some rational values, called allowed values. We find expressions of <i>a</i> and <i>s</i> as a function of <i>M</i> for the allowed values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and <i>M</i> and parametric expressions of <i>a</i>, <i>M</i>, and <i>s</i>. Further, Pell equations deduced from the conditions of <i>M</i> are solved to find the allowed values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and to provide all solutions in <i>a</i> and <i>M</i>. These results yield new insights into the overall properties of the classical problem of the sums of consecutive squared integers equal to squared integers and allow us to solve this problem completely by providing all solutions in infinite families. |
| first_indexed | 2024-03-07T22:12:24Z |
| format | Article |
| id | doaj.art-c4823accf5a54a269b927828006de640 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-07T22:12:24Z |
| publishDate | 2024-01-01 |
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| series | Symmetry |
| spelling | doaj.art-c4823accf5a54a269b927828006de6402024-02-23T15:35:50ZengMDPI AGSymmetry2073-89942024-01-0116214610.3390/sym16020146Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared IntegersVladimir Pletser0European Space Research and Technology Centre, European Space Agency, 2201 AZ Noordwijk, The NetherlandsThe classical problem of finding all integers <i>a</i> and <i>M</i> such that the sums of <i>M</i> consecutive squared integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mfenced separators="" open="(" close=")"><mi>a</mi><mo>+</mo><mi>i</mi></mfenced><mn>2</mn></msup></semantics></math></inline-formula> equal the squared integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>s</mi><mn>2</mn></msup></semantics></math></inline-formula>, where <i>M</i> is the number of terms in the sum, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>a</mi><mn>2</mn></msup></semantics></math></inline-formula> the first term and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>M</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, yields remarkable regular linear features when plotting the values of <i>M</i> as a function of <i>a</i>. These linear features correspond to groupings of pairs of <i>a</i> values for successive same values of <i>M</i> found on either side of straight lines of equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mi>M</mi><mo>=</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula>, where <i>c</i> is an integer constant and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> a parameter taking some rational values, called allowed values. We find expressions of <i>a</i> and <i>s</i> as a function of <i>M</i> for the allowed values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and <i>M</i> and parametric expressions of <i>a</i>, <i>M</i>, and <i>s</i>. Further, Pell equations deduced from the conditions of <i>M</i> are solved to find the allowed values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and to provide all solutions in <i>a</i> and <i>M</i>. These results yield new insights into the overall properties of the classical problem of the sums of consecutive squared integers equal to squared integers and allow us to solve this problem completely by providing all solutions in infinite families.https://www.mdpi.com/2073-8994/16/2/146sums of consecutive squared integers equal to square integersquadratic Diophantine equationgeneralized Pell equationfundamental solutionsChebyshev polynomials |
| spellingShingle | Vladimir Pletser Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers Symmetry sums of consecutive squared integers equal to square integers quadratic Diophantine equation generalized Pell equation fundamental solutions Chebyshev polynomials |
| title | Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers |
| title_full | Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers |
| title_fullStr | Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers |
| title_full_unstemmed | Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers |
| title_short | Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers |
| title_sort | linear relations between numbers of terms and first terms of sums of consecutive squared integers equal to squared integers |
| topic | sums of consecutive squared integers equal to square integers quadratic Diophantine equation generalized Pell equation fundamental solutions Chebyshev polynomials |
| url | https://www.mdpi.com/2073-8994/16/2/146 |
| work_keys_str_mv | AT vladimirpletser linearrelationsbetweennumbersoftermsandfirsttermsofsumsofconsecutivesquaredintegersequaltosquaredintegers |