Kornachchi Algorithm

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

Content

Algorithm

First we find any solution $r_{0}^{2}\equiv -d{\pmod {m}}$ ${\ displaystyle r_ {0} ^ {2} \ equiv -d {\ pmod {m}}}$ . If such $r_{0}$ ${\ displaystyle r_ {0}}$ does not exist, the original equation has no primitive solutions. Without loss of generality, we can assume that $r_{0}\leq {\tfrac {m}{2}}$ ${\ displaystyle r_ {0} \ leq {\ tfrac {m} {2}}}$ (if this is not so, replace r ₀ with m - r ₀ , which remains the root of - d ). Now we use the Euclidean algorithm to search $r_{1}\equiv m{\pmod {r_{0}}}$ ${\ displaystyle r_ {1} \ equiv m {\ pmod {r_ {0}}}}$ , $r_{2}\equiv r_{0}{\pmod {r_{1}}}$ ${\ displaystyle r_ {2} \ equiv r_ {0} {\ pmod {r_ {1}}}}$ and so on. Stop when $r_{k}<{\sqrt {m}}$ ${\ displaystyle r_ {k} <{\ sqrt {m}}}$ . If a $s={\sqrt {\tfrac {m-r_{k}^{2}}{d}}}$ ${\ displaystyle s = {\ sqrt {\ tfrac {m-r_ {k} ^ {2}} {d}}}}$ is an integer, then the solution will be $x=r_{k},y=s$ ${\ displaystyle x = r_ {k}, y = s}$ . Otherwise, there is no primitive solution.

To search for non-primitive solutions ( x , y ), where GCD ( x , y ) = g ≠ 1, we note from the existence of such a solution that g ² divides m (and, equivalently, if m is square-free , then all solutions are primitive). Then the above algorithm can be used to search for a primitive solution ( u , v ) of the equation $u^{2}+dv^{2}={\tfrac {m}{g^{2}}}$ ${\ displaystyle u ^ {2} + dv ^ {2} = {\ tfrac {m} {g ^ {2}}}}$ . If such a solution is found, then ( gu , gv ) will be a solution to the original equation.

Example

We solve the equation $x^{2}+6y^{2}=103$ ${\ displaystyle x ^ {2} + 6y ^ {2} = 103}$ . The square root of −6 (mod 103) is 32 and 103 ≡ 7 (mod 32). Insofar as $7^{2}<103$ ${\ displaystyle 7 ^ {2} <103}$ and ${\sqrt {\tfrac {103-7^{2}}{6}}}=3$ ${\ displaystyle {\ sqrt {\ tfrac {103-7 ^ {2}} {6}}} = 3}$ , there exists a solution x = 7, y = 3.

Notes

↑ Cornacchia, 1908 , p. 33–90.

Literature

G. Cornacchia. Su di un metodo per la risoluzione in numeri interi dell 'equazione $\sum _{h=0}^{n}C_{h}x^{n-h}y^{h}=P$ ${\ displaystyle \ sum _ {h = 0} ^ {n} C_ {h} x ^ {nh} y ^ {h} = P}$ . // Giornale di Matematiche di Battaglini. - 1908. - T. 46 .

Links

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

[_2a777945ddc9fb54-1] Cornacchia, 1908 , p. 33–90.