JP Journal of Algebra, Number Theory and Applications
Volume 38, Issue 6, Pages 647 - 664
(December 2016) http://dx.doi.org/10.17654/NT038060647 |
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ON THE NUMBER OF DISTINCT NON-VANISHING REPRESENTATIONS OF PRIME POWERS BY SUMS OF THREE AND FOUR SQUARES
Werner Hürlimann
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Abstract: We derive closed-form counting formulas for distinct non-vanishing representations of prime powers by sums of three and four squares. The formulas for even powers of primes depend only on the residue classes of primes modulo 8 (modulo 24) for three squares (four squares). The formulas for odd powers of primes depend via the theorem of Gauss on class numbers of binary quadratic forms with negative discriminants. |
Keywords and phrases: sum of squares, prime powers, Pythagorean quadruple, Pythagorean quintuple, theorem of Gauss, class number, counting formula. |
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