Cornacchia's algorithm

In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation $x^2+d{y}^2=m$, where $1\le d<m$ and d and m are coprime. The algorithm was described in 1908 by Giuseppe Cornacchia.

The algorithm

First, find any solution to $r_0^2\equiv -d(\mathrm{mod}m)$ (perhaps by using an algorithm listed here); if no such $r_0$ exist, there can be no primitive solution to the original equation. Without loss of generality, we can assume that r₀ ≤ m/2 (if not, then replace r₀ with m - r₀, which will still be a root of -d). Then use the Euclidean algorithm to find $r_1\equiv m(\mathrm{mod}{r}_0)$, $r_2\equiv r_0(\mathrm{mod}{r}_1)$ and so on; stop when $r_k<\sqrt{m}$. If $s=\sqrt{\frac{m-r_k^2}{d}}$ is an integer, then the solution is $x=r_k,y=s$; otherwise try another root of -d until either a solution is found or all roots have been exhausted. In this case there is no primitive solution.

To find non-primitive solutions (x, y) where gcd(x, y) = g ≠ 1, note that the existence of such a solution implies that g² divides m (and equivalently, that if m is square-free, then all solutions are primitive). Thus the above algorithm can be used to search for a primitive solution (u, v) to u² + dv² = m/g². If such a solution is found, then (gu, gv) will be a solution to the original equation.

Example

Solve the equation $x^2+6y^2=103$. A square root of −6 (mod 103) is 32, and 103 ≡ 7 (mod 32); since $7^2<103$ and $\sqrt{\frac{103-7^2}{6}}=3$, there is a solution x = 7, y = 3.

External links

• Basilla, J. M. (2004). "On the solutions of $x^2+d{y}^2=m$" (PDF). Proc. Japan Acad. 80(A): 40-41. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1116442240.