Cornacchia's algorithm

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

The algorithm

First, find any solution to $r_{0}^{2}\equiv -d{\pmod m}$ (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 $r 0 \leq m / 2$ (if not, then replace $r 0$ with $m - r 0$ , which will still be a root of $- d$ ). 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 primitive solution.

To find non-primitive solutions $(x, y)$ where $gcd(x, y) = g \neq 1$ , note that the existence of such a solution implies that $g 2$ 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 $u 2 + dv 2 = m / g 2$ . 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 ${\sqrt {{\tfrac {103-7^{2}}6}}}=3$ , there is a solution x = 7, y = 3.

References

↑ Cornacchia, G. (1908). "Su di un metodo per la risoluzione in numeri interi dell' equazione $\sum _{{h=0}}^{n}C_{h}x^{{n-h}}y^{h}=P$ .". 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 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 LLL Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms

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