Abstract
An integern is congruent if there is a triangle with rational sides whose area isn. 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 useO(n) operations. In this paper we give several methods with smaller exponents, including a randomized algorithm using timen 11/2+o(1) and spacen o(1) and a deterministic algorithm using space and timen 1/2+o(1).
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