# Conclusion

Part of the CMS Books in Mathematics book series (CMSBM)
In this section1 we will investigate the Diophantine equation
$$ax^2 + bxy + cy^2 + dx + ey + f = 0\,\,,$$
(17.1)
, where it is required to find integral values of x and y, given a, b, c, d, e, f$$\mathbb{Z}$$. This is the most general form of a quadratic Diophantine equation in two variables and, as such, represents a further generalization of the Pell equation.2 A method for solving this equation was given over 200 years ago by Lagrange,3 and this method has not been improved significantly since that time. The reason for this is that Lagrange’s method works perfectly well as long as the coefficients in (17.1) do not get very large. However, if we put H = max{¦a¦, ¦b¦, ¦c¦, ¦d¦, ¦e¦, ¦f¦}, it has been shown4 that there is an infinite collection of equations of the form (17.1) having integer solutions x, y, but none with max{¦x¦, ¦y¦} ≤ 2H/5. Thus, it is possible for solutions of (17.1) to be very large, even when H is only moderately large. For such cases, Lagrange’s method will likely be far too slow to produce the solutions of (17.1). We will show here how our previously developed results can be used to produce a faster method for solving this equation.

## Keywords

Diophantine Equation Hyperelliptic Curve Elliptic Curve Cryptography Discrete Logarithm Problem Algebraic Integer