# 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} }

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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