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.
References
[1]
Stillwell, J. (1998) Numbers and Geometry. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-0687-3
[2]
Stillwell, J. (2003) Elements of Number Theory. Springer-Verlag, New York. https://doi.org/10.1007/978-0-387-21735-2
[3]
Rademacher, H. and Toeplitz, O. (2023) The Enjoyment of Math. Princeton University Press, Princeton.
[4]
Stillwell, J. (2002) Mathematics and Its History. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4684-9281-1
[5]
Pickover, C.A. (2009) The Mαth βook. Sterling, New York.
[6]
Beiler, A.H. (1966) Recreations in the Theory of Numbers. Dover Publications, New York.
[7]
Singh, S. (1997) Fermat’s Last Theorem. GB Gardners Books, Eastbourne, UK.
[8]
Conway, J.H. and Smith, D.A. (2003) On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. CRC Press, New York. https://doi.org/10.1201/9781439864180
[9]
Gürlebeck, K. and Sprössig, W. (1997) Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons, New York.
[10]
Koecher, M. and Remmert, R. (1991) Hamilton’s Quaternions. In: Ebbinghaus, H.D., Ed., Numbers, Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-1005-4_10
[11]
Ebbinghaus, H.D., et al. (1991) Numbers. Springer-Verlag, New York.
