Abstract
The general binary quadratic Diophantine equation
was first solved by Lagrange over 200 years ago. Since that time little improvement has been made to Lagrange’s technique. In this paper we show how to reduce this problem to that of determining whether or not an ideal of a certain quadratic order is principal and if so exhibiting a generator of that ideal. In the difficult case of the discriminant \(\ensuremath{\Delta}\) of this order being positive, we develop a Las Vegas algorithm for solving the principal ideal problem that executes in expected time bounded by \(O(\ensuremath{\Delta}^{1/6 + \epsilon})\), whereas the complexity of Lagrange’s (unconditional) technique for solving this problem is \(O(\ensuremath{\Delta}^{1/2 + \epsilon})\).
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Sawilla, R.E., Silvester, A.K., Williams, H.C. (2008). A New Look at an Old Equation. In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_2
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