Author(s)

Alfred Wünsche

Institute of Physics, Humboldt University, Berlin, Germany.

The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a²+b²=c² with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a²+b²+c²=d² with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d²<1000 is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a²+b²+c²+d²=e² and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d²<1000, respectively, e²<500.

Number Theory, Pythagorean Triples, Tesseract, 4-Square Identity, Diophantine Equation

Wünsche, A. (2024) Three- and Four-Dimensional Generalized Pythagorean Numbers. Advances in Pure Mathematics, 14, 1-15. doi: 10.4236/apm.2024.141001.