TITLE:
Representation of an Integer by a Quadratic Form through the Cornacchia Algorithm
AUTHORS:
Moumouni Djassibo Woba
KEYWORDS:
Quadratic Form, Cornacchia Algorithm, Associated Polynomials, Euclid’s Algorithm, Prime Number
JOURNAL NAME:
Applied Mathematics,
Vol.15 No.9,
September
12,
2024
ABSTRACT: Cornachia’s algorithm can be adapted to the case of the equation
x
2
+d
y
2
=n
and even to the case of
a
x
2
+bxy+c
y
2
=n
. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation
x
2
+
y
2
=n
). Starting from a quadratic form with two variables
f(
x,y
)=a
x
2
+bxy+c
y
2
and n an integer. We have shown that a primitive positive solution
(
u,v
)
of the equation
f(
x,y
)=n
is admissible if it is obtained in the following way: we take α modulo n such that
f(
α,1
)≡0modn
, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than
4cn/
| D |
) (possibly α itself) and the equation
f(
x,y
)=n
. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form
n=a
x
2
+bxy+c
y
2
if all the primitive positive integer solutions of the equation
f(
x,y
)=n
are admissible, i.e. computable by the algorithmic process.