Open Access
October 2007 Efficient Verification of Tunnell's Criterion
Eric Bach, Nathan C. Ryan
Japan J. Indust. Appl. Math. 24(3): 229-239 (October 2007).


An integer $n$ is congruent if there is a triangle with rational sides whose area is $n$. In the 1980s Tunnell gave an algorithm to test congruence which relied on counting integral points on the ellipsoids $2x^2+y^2+8z^2 = n$ and $2x^2+y^2+32z^2=n$. The correctness of this algorithm is conditional on the conjecture of Birch and Swinnerton-Dyer. The known methods for testing Tunnell's criterion use $O(n)$ operations. In this paper we give several methods with smaller exponents, including a randomized algorithm using time $n^{1/2 + o(1)}$ and space $n^{o(1)}$, and a deterministic algorithm using space and time $n^{1/2 + o(1)}$.


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Eric Bach. Nathan C. Ryan. "Efficient Verification of Tunnell's Criterion." Japan J. Indust. Appl. Math. 24 (3) 229 - 239, October 2007.


Published: October 2007
First available in Project Euclid: 17 December 2007

zbMATH: 1183.11015
MathSciNet: MR2374988

Keywords: algorithms , class field theory , Complexity , congruent numbers , Quadratic forms

Rights: Copyright © 2007 The Japan Society for Industrial and Applied Mathematics

Vol.24 • No. 3 • October 2007
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