On representations of a number as a sum of three squares

Michael D. Hirschhorn, James A. Sellers

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22 Scopus citations


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 languageEnglish (US)
Pages (from-to)85-101
Number of pages17
JournalDiscrete Mathematics
Issue number1-3
StatePublished - Mar 28 1999

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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