### Unisequences and nearest integer continued fraction midpoint criteria for Pell's equation.

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We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $a{x}^{2}+bxy+c{y}^{2}=N$ in relatively prime integers $x,y$, where $N\ne 0$, gcd$(a,b,c)=\text{gcd}(a,N)=1\phantom{\rule{4.0pt}{0ex}}\text{et}\phantom{\rule{4.0pt}{0ex}}D={b}^{2}-4ac\>0$ is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation ${x}^{2}-D{y}^{2}=N$. As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s for...

We present an easy-to-implement algorithm for transforming a matrix to rational canonical form.

We show that, with suitable modification, the upper bound estimates of Stolt for the fundamental integer solutions of the Diophantine equation Au²+Buv+Cv²=N, where A>0, N≠0 and B²-4AC is positive and nonsquare, in fact characterize the fundamental solutions. As a corollary, we get a corresponding result for the equation u²-dv²=N, where d is positive and nonsquare, in which case the upper bound estimates were obtained by Nagell and Chebyshev.

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