%0 Journal Article
%T Three- and Four-Dimensional Generalized Pythagorean Numbers
%A Alfred W¨šnsche
%J Advances in Pure Mathematics
%P 1-15
%@ 2160-0384
%D 2024
%I Scientific Research Publishing
%R 10.4236/apm.2024.141001
%X The Pythagorean triples (<em>a, b </em>| <em>c</em>)<em></em> of planar geometry which satisfy the equation <em>a</em><sup><em>2</em></sup><em>+b</em><sup><em>2</em></sup><em>=c</em><sup><em>2</em></sup> with integers (<em>a, b, c</em>) are generalized to 3D-Pythagorean quadruples (<em>a, b, c </em>| <em>d</em>) of spatial geometry which satisfy the equation <em>a</em><sup><em>2</em></sup><em>+b</em><sup><em>2</em></sup><em>+c</em><sup><em>2</em></sup><em>=d</em><sup><em>2</em></sup> with integers <em>(a, b, c, d)</em>. Rules for a parametrization of the numbers (<em>a, b, c, d</em>)<em></em><em></em> are derived and a list of all possible nonequivalent cases without common divisors up to<em> d</em><sup><em>2</em></sup><em>&lt;1000</em> is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (<em>a, b, c, d </em>| <em>e</em>) <em></em>which satisfy the equation <em>a</em><sup><em>2</em></sup><em>+b</em><sup><em>2</em></sup><em>+c</em><sup><em>2</em></sup><em>+d</em><sup><em>2</em></sup><em>=e</em><sup><em>2</em></sup> and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the <em>N</em>-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to <em>d</em><sup><em>2</em></sup><em>&lt;1000</em>, respectively, <em>e</em><sup><em>2</em></sup><em>&lt;500</em>.
%K Number Theory
%K Pythagorean Triples
%K Tesseract
%K 4-Square Identity
%K Diophantine Equation
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=130346