Cornacchia's algorithm

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

The algorithm

First, find any solution to ${\displaystyle r_{0}^{2}\equiv -d{\pmod {m}}}$ (perhaps by using an algorithm listed here); if no such ${\displaystyle r_{0}}$ 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 ${\displaystyle r_{1}\equiv m{\pmod {r_{0}}}}$, ${\displaystyle r_{2}\equiv r_{0}{\pmod {r_{1}}}}$ and so on; stop when ${\displaystyle r_{k}<{\sqrt {m}}}$. If ${\displaystyle s={\sqrt {\tfrac {m-r_{k}^{2}}{d}}}}$ is an integer, then the solution is ${\displaystyle x=r_{k},y=s}$; 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 ${\displaystyle x^{2}+6y^{2}=103}$. A square root of −6 (mod 103) is 32, and 103 ≡ 7 (mod 32); since ${\displaystyle 7^{2}<103}$ and ${\displaystyle {\sqrt {\tfrac {103-7^{2}}{6}}}=3}$, there is a solution x = 7, y = 3.

References

1. Cornacchia, G. (1908). "Su di un metodo per la risoluzione in numeri interi dell' equazione ${\displaystyle \sum _{h=0}^{n}C_{h}x^{n-h}y^{h}=P}$". Giornale di Matematiche di Battaglini. 46: 33–90.

External links

• Basilla, J. M. (2004). "On the solutions of ${\displaystyle x^{2}+dy^{2}=m}$" (PDF). Proc. Japan Acad. 80(A): 40–41.