Japan Journal of Industrial and Applied Mathematics

Efficient Verification of Tunnell's Criterion

Eric Bach and Nathan C. Ryan

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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 $n^{1/2 + o(1)}$.

Article information

Japan J. Indust. Appl. Math. Volume 24, Number 3 (2007), 229-239.

First available in Project Euclid: 17 December 2007

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congruent numbers quadratic forms class field theory algorithms complexity


Bach, Eric; Ryan, Nathan C. Efficient Verification of Tunnell's Criterion. Japan J. Indust. Appl. Math. 24 (2007), no. 3, 229--239. https://projecteuclid.org/euclid.jjiam/1197909110.

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