A simple derivation of Jacobi’s four-square formula

Ewell, John A.

doi:10.1090/S0002-9939-1982-0656093-9

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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A simple derivation of Jacobi’s four-square formula
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by John A. Ewell PDF

Proc. Amer. Math. Soc. 85 (1982), 323-326 Request permission

Abstract:

For each nonnegative integer $n$, ${r_4}(n)$ counts number of solutions $({x_1},{x_2},{x_3},{x_4}) \in {{\mathbf {Z}}^4}{\text {of }}n = x_1^2 + x_2^2 + x_3^2 + x_4^2$. Within the confines of elementary partition theory the author presents a simple derivation of Jacobi’s formula for ${r_4}(n)$.

References

History of the theory of numbers

John A. Ewell, Completion of a Gaussian derivation, Proc. Amer. Math. Soc. 84 (1982), no. 2, 311–314. MR 637190, DOI 10.1090/S0002-9939-1982-0637190-0

An introduction to the theory of numbers