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Mathematics of Computation

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Dirichlet's proof of the three-square theorem: An algorithmic perspective

Authors: Paul Pollack and Peter Schorn
Journal: Math. Comp. 88 (2019), 1007-1019
MSC (2010): Primary 11E25; Secondary 11Y50
Published electronically: May 29, 2018
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Abstract: The Gauss-Legendre three-square theorem asserts that the positive integers $ n$ expressible as a sum of three integer squares are precisely those not of the form $ 4^k(8m+7)$ for any nonnegative integers $ k,m$. In 1850, Dirichlet gave a beautifully simple proof of this result using only basic facts about ternary quadratic forms. We explain how to turn Dirichlet's proof into an algorithm; if one assumes the Extended Riemann Hypothesis (ERH), there is a random algorithm for expressing $ n=x^2+y^2+z^2$, where the expected number of bit operations is $ O((\lg n)^2 (\lg \lg n)^{-1} \cdot M(\lg n))$. Here, $ M(r)$ stands in for the bit complexity of multiplying two $ r$-bit integers. A random algorithm for this problem of similar complexity was proposed by Rabin and Shallit in 1986; however, their analysis depended on both the ERH and on certain conjectures of Hardy-Littlewood type.

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Additional Information

Paul Pollack
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Peter Schorn
Affiliation: Culmannstrasse 77, CH-8006 Zurich, Switzerland

Received by editor(s): July 16, 2017
Received by editor(s) in revised form: July 18, 2017, and December 4, 2017
Published electronically: May 29, 2018
Additional Notes: The research of the first-named author was supported by NSF award DMS-1402268.
Article copyright: © Copyright 2018 American Mathematical Society

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