INFINITE FAMILIES OF ARITHMETIC IDENTITIES FOR 4-CORES

Publisher： Cambridge University Press

E-ISSN： 1755-1633|87|2|304-315

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.87, Iss.2, 2013-04, pp. : 304-315

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Abstract

Let u(n) and v(n) be the number of representations of a nonnegative integer n in the forms x 2+4y 2+4z 2 and x 2+2y 2+2z 2, respectively, with x,y,z∈ℤ, and let a 4(n) and r 3(n) be the number of 4-cores of n and the number of representations of n as a sum of three squares, respectively. By employing simple theta-function identities of Ramanujan, we prove that $u(8n+5)=8a_4(n)=v(8n+5)=\frac {1}{3}r_3(8n+5)$ . With the help of this and a classical result of Gauss, we find a simple proof of a result on a 4 (n) proved earlier by K. Ono and L. Sze [‘4-core partitions and class numbers’, Acta Arith. 80 (1997), 249–272]. We also find some new infinite families of arithmetic relations involving a 4 (n) .

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