Dirichlet’s proof of the three-square theorem: An algorithmic perspective
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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.References
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Additional Information
- Paul Pollack
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 830585
- Email: pollack@uga.edu
- Peter Schorn
- Affiliation: Culmannstrasse 77, CH-8006 Zurich, Switzerland
- MR Author ID: 249112
- Email: peter.schorn@acm.org
- 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.
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1007-1019
- MSC (2010): Primary 11E25; Secondary 11Y50
- DOI: https://doi.org/10.1090/mcom/3349
- MathSciNet review: 3882293