# Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem

@article{Hallgren2007PolynomialtimeQA,
title={Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem},
author={Sean Hallgren},
journal={J. ACM},
year={2007},
volume={54},
pages={4:1-4:19}
}
• Sean Hallgren
• Published 1 March 2007
• Mathematics, Computer Science
• J. ACM
We give polynomial-time quantum algorithms for three problems from computational algebraic number theory. The first is Pell's equation. Given a positive nonsquare integer d, Pell's equation is x2 − dy2 = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem we solve is the principal ideal problem in real quadratic number fields…
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