# On representations of a number as a sum of three squares

Michael D. Hirschhorn, James A. Sellers

Research output: Contribution to journalArticle

19 Citations (Scopus)

### Abstract

We give a variety of results involving s(n), the number of representations of n as a sum of three squares. Using elementary techniques we prove that if 9 + n, s(9λn) = (3λ+1-1/2-(-n/3)3λ+1-1/2)s(n); via the theory of modular forms of half integer weight we prove the corresponding result with 3 replaced by p, an odd prime. This leads to a formula for s(n) in terms of s(n′), where n′ is the square-free part of n. We also find generating function formulae for various subsequences of {s(n)}, for instance Σs(3n + 2)qn = 12 Π (1 + q2n-1)2(1 - q6n)3. n≥0 n≥1

Original language English (US) 85-101 17 Discrete Mathematics 199 1-3 Published - Mar 28 1999

### Fingerprint

Subsequence
Square free
Modular Forms
Generating Function
Odd
Integer

### All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Discrete Mathematics and Combinatorics

### Cite this

Hirschhorn, M. D., & Sellers, J. A. (1999). On representations of a number as a sum of three squares. Discrete Mathematics, 199(1-3), 85-101.
Hirschhorn, Michael D. ; Sellers, James A. / On representations of a number as a sum of three squares. In: Discrete Mathematics. 1999 ; Vol. 199, No. 1-3. pp. 85-101.
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Hirschhorn, MD & Sellers, JA 1999, 'On representations of a number as a sum of three squares', Discrete Mathematics, vol. 199, no. 1-3, pp. 85-101.

On representations of a number as a sum of three squares. / Hirschhorn, Michael D.; Sellers, James A.

In: Discrete Mathematics, Vol. 199, No. 1-3, 28.03.1999, p. 85-101.

Research output: Contribution to journalArticle

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Hirschhorn MD, Sellers JA. On representations of a number as a sum of three squares. Discrete Mathematics. 1999 Mar 28;199(1-3):85-101.