| In this paper, we use the theory of elliptic function to prove a theta function identity. Using this identity, we re-derive the representations of η2(τ), η4(τ), η6(τ), η8(τ) and η10(τ) as well as some theta function addition-formulaes due to Jacobi. With the help of these addition-formulaes, we get formulas for the numbers of representations of a given natural number by sums of two , four and eight square numbers.
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