Expressing (a^2+b^2+c^2+d^2)^3 as a Sum of 23 Sixth Powers(*)

R. H. Hardin  and  N. J. A. Sloane(**)

Mathematical Sciences Research Center 
AT&T Bell Laboratories, Murray Hill, NJ 07974 

(**) Present address:
Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971 USA
Email: njas@research.att.com


ABSTRACT

It is shown that (x_1^2 + x_2^2 +x_3^2 + x_4^2)^3 can be written
as a sum of 23 sixth powers of linear forms.
This is one less than is required in Kempner's 1912 identity.
There is a corresponding set of 23 points in the four-dimensional unit ball
which provide an exact quadrature rule for homogeneous polynomials
of degree 6 on S^3.
It appears that this result is best possible, i.e. that no 22-term identity exists.

(*) A different version of this paper appeared in Journal of Combinatorial Theory,
Series A, 68 (1994), 481-485. 

Mathematics Subject Classification:
Primary 11P05 (Waring's problem); secondary 65D32 (quadrature formulas). 

Key words and phrases: Waring's problem,
isometric embeddings, quadrature formulas for sphere


For the full version see
http://NeilSloane.com/doc/lucas.pdf (pdf) or
http://NeilSloane.com/doc/lucas.ps (ps)