TY - JOUR

T1 - On representations of a number as a sum of three squares

AU - Hirschhorn, Michael D.

AU - Sellers, James A.

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1999/3/28

Y1 - 1999/3/28

N2 - 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

AB - 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

KW - Generating functions

KW - Jacobi's triple product

KW - Representations

KW - Three squares

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U2 - 10.1016/s0012-365x(98)00288-x

DO - 10.1016/s0012-365x(98)00288-x

M3 - Article

AN - SCOPUS:0043209368

VL - 199

SP - 85

EP - 101

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -