Cornacchia's algorithm

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

1 The algorithm
2 Example
3 References
4 External links

The algorithm

First, find any solution to $r_0^2\equiv-d\pmod m$ ; if no such $r_0$ exist, there can be no primitive 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 $x=r_k,y=s$ ; otherwise there is no solution.

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 $\sqrt{\tfrac{103-7^2}6}=3$ , there is a solution x = 7, y = 3.

References

^ 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.

External links

Basilla, Julius Magalona (12 May 2004). "On Cornacchia's algorithm for solving the diophantine equation $u^2+dv^2=m$ " (PDF).

Number-theoretic algorithms

Primality tests	AKS test APR test Baillie–PSW ECPP test Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin

Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization

Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's

Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's

Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve

Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's

Modular square root	Cipolla Pocklington's Tonelli–Shanks

Other algorithms	Chakravala Cornacchia Integer relation Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms Smallcaps indicate a deterministic algorithm

fonte: https://web.archive.org/web/20150816172325/https://en.wikipedia.org/wiki/Cornacchia%27s_algorithm

Conheça Walt Disney World