Here :
http://mathworld.wolfram.com/PellEquation.html
the pell-like equation $$y^2-Dx^2=c$$ is mentioned. It is claimed that in the case $|c|<\sqrt{D}$, a solution must occur among the convergents, if I understand it right.
For the equation $$y^2-2473x^2=44$$ , this is satisfied. However, I calculated a solution of $11x^2+47xy-6y^2=44$ by using the convergents of one of the roots of $11x^2+47x-6$ and found out that
$$x=794538928954973566949076$$ $$y=15977290161340977278578$$
solves the given equation and we have $44<\sqrt{2473}$
But I did not find a convergent $\frac{p_n}{q_n}$ of $\sqrt{2473}$ satisfying $p_n^2-2473q_n^2=44$
What is wrong with the claim ? Or where is my mistake ?
Sticking with Pell type, the search for $x^2 - n y^2 = c$ is not infinite. If there is any integer solution that is primitive, meaning $\gcd(x,y) = 1,$ then the automorphism group of the form $x^2 - n y^2$ takes us to solutions that obey certain inequalities. I wrote a program based on this... note $7 \cdot 17 \cdot 23 = 2737,$ and there are eight orbits under the automorphism group.
In this case, a solution $x,y > 0,$ $x^2 - 2 y^2 = 2737$ is called a "seed" when either $$ 3x - 4 y \leq 0 \; \; \; \mbox{OR} \; \; \; -2x+3y \leq 0. $$ A solution with $x,y >0$ that is not a seed allows for a smaller positive solution (in the same orbit) $$ (3x-4y, -2x+3y) $$
If you draw some careful pictures of the lines and hyperbola involved in the inequalities, you see how this leads to bounds, which can be made explicit. As usual for Pell type, one of the inequalities (blue) is irrelevant, so there is just the blue line in the graph, no shading.
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Notice how the other inequality becomes relevant when we switch to $x^2 - 2 y^2 = -2737.$
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