| Summary: | 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.
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