Three- and Four-Dimensional Generalized Pythagorean Numbers
- 1 Institute of Physics, Humboldt University, Berlin, Germany
Abstract
The Pythagorean triples ( a, b | c ) of planar geometry which satisfy the equation a 2 +b 2 =c 2 with integers ( a, b, c ) are generalized to 3D-Pythagorean quadruples ( a, b, c | d ) of spatial geometry which satisfy the equation a 2 +b 2 +c 2 =d 2 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 d 2 <1000 is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples ( a, b, c, d | e ) which satisfy the equation a 2 +b 2 +c 2 +d 2 =e 2 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 d 2 <1000 , respectively, e 2 <500 .
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