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A simple derivation of Jacobi's four-square formula


Author: John A. Ewell
Journal: Proc. Amer. Math. Soc. 85 (1982), 323-326
MSC: Primary 10J05; Secondary 10A45
DOI: https://doi.org/10.1090/S0002-9939-1982-0656093-9
MathSciNet review: 656093
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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 [Enhancements On Off] (What's this?)

  • [1] L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York, 1952.
  • [2] J. A. Ewell, Completion of a Gaussian derivation, Proc. Amer. Math. Soc. 84 (1982), 311-314. MR 637190 (83a:10084)
  • [3] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Clarendon Press, Oxford, 1960.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0656093-9
Keywords: Four-square theorem, Jacobi's formula for $ {r_4}(n)$
Article copyright: © Copyright 1982 American Mathematical Society

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