In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation
and d and m are coprime.
The algorithm was described in 1908 by Giuseppe Cornacchia.
[1] First, find any solution to
( mod
(perhaps by using an algorithm listed here); if no such
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
( mod
( mod
is an integer, then the solution is
; 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.
Solve the equation
A square root of −6 (mod 103) is 32, and 103 ≡ 7 (mod 32); since