## DIMACS TR: 93-54

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

### Authors: R. H. Hardin and N. J. A. Sloane

**
ABSTRACT
**

It is shown that
(x sup 1 sup 2 + x sub 2 sup 2 +x sub 3 sup 2 + x sub 4 sup 2 ) sup 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 sup 3 . It appears that this result is best possible,
i.e. that no 22-term identity exists.

Paper available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-54.ps

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