# Cornacchia's algorithm

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Cornacchia's algorithm

In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation x2 + dy2 = m, where $1\le d and d and m are coprime. The algorithm was described in 1908 by Giuseppe Cornacchia[1].

## The algorithm

First, find any solution to $r_0^2\equiv-d\pmod m$; if no such r0 exist, there can be no solution to the original equation. Then use the Euclidean algorithm to find $r_1\equiv m\pmod{r_0}$, $r_2\equiv r_0\pmod{r_1}$ and so on; stop when $r_k<\sqrt m$. If $s=\sqrt{\tfrac{m-r_k^2}d}$ is an integer, then the solution is x = rk,y = s; otherwise there is no solution.

## Example

Solve the equation x2 + 6y2 = 103. A square root of −6 (mod 103) is 32, and 103 ≡ 7 (mod 32); since 72 < 103 and $\sqrt{\tfrac{103-7^2}6}=3$, 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 $\sum_{h=0}^nC_hx^{n-h}y^h=P$.". Giornale di Matematiche di Battaglini 46: 33–90.

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