Quantum Algorithm for Solving Pell’s Equation

Years and Authors of Summarized Original Work

• 2002; Hallgren

Problem Definition

Pell’s equation is one of the oldest studied problem in number theory. For a positive square-free integer d, Pell’s equation is $x^{2} -\mathit{dy}^{2} = 1$, and the problem is to compute integer solutions x, y of the equation [8, 10]. The earliest algorithm for it uses the continued fraction expansion of $\sqrt{ d}$ and dates back to 1000 a.d. by Indian mathematicians. Lagrange showed that there are an infinite number of solutions of Pell’s equation. All solutions are of the form $x_{n} + y_{n}\sqrt{d} = (x_{1} + y_{1}\sqrt{d})^{n}$, where the smallest solution, $(x_{1},y_{1})$, is called the fundamental solution. The solution $(x_{1},y_{1})$ may have exponentially many bits in general in terms of the input size, which is $\log d$, and so cannot be written down in polynomial time. To resolve this difficulty, the computational problem is recast as computing the integer closest to the regulator \(...