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A simple proof of Jacobi’s two-square theorem. (English) Zbl 0577.10041

In a recent note, J. A. Ewell [ibid. 90, 635-637 (1983; Zbl 0526.10038)] derives Fermat’s two-square theorem: ”A prime \(p=4n+1\) is the sum of two squares” from the triple-product identity. Here the author observes that from the triple-product identity one can obtain a stronger result due to Jacobi, and proves the following Theorem. The number \(r_ 2(n)\) of representations of the positive integer n as a sum of two squares is given by \(r_ 2(n)=4(d_ 1(n)-d_ 3(n))\), where \(d_ i(n)= \sum_{d| n, d\equiv i mod 4}1\).

MSC:

11P05 Waring’s problem and variants

Citations:

Zbl 0526.10038
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