Why does $29^2 : 31^2 : 41^2$ have a close integer approximation with small numbers? "Everybody knows" that such coincidences as
$$2\times2\times\overbrace{41\times41} = 6724 \approx 6728 = 2\times2\times2\times\overbrace{29\times29}$$
(And why did I bother with the first two factors of $2$ on each side?  Be patient.)
are "explained" by the fact that $\dfrac{41}{29}$ is a convergent in the simple continued fraction expansion of $\sqrt 2$. and maybe
$$2\times2\times2\times\overbrace{29\times29} = 6728 \approx 6727 = 7\times\overbrace{31\times31}$$
has a similar "explanation", as presumably would the fact that
$$2\times2\times\overbrace{41\times41} = 6724 \approx 6727 = 7\times\overbrace{31\times31}.$$
Is there some such "explanation" of the simultaneous proximity of all three of these numbers to each other?
 A: I think it's just the law of small numbers. I'm assuming you want distinct triples $x_1,x_2,x_3$ such that $x_1\approx x_2\approx x_3$ as well as $ax_1^2\approx bx_2^2\approx cx_3^2$ with distinct $a,b,c$. If so, yours was the first of several examples and Mathematica quickly finds,
$$\begin{aligned}
6724&= 4\times41^2\\
6727&= 7\times31^2\\
6728&= 8\times29^2\\
\end{aligned}\tag1$$
$$\begin{aligned}
7935 &= 15\times23^2\\
7938 &= 18\times21^2\\
7942 &= 22\times19^2\\
\end{aligned}\tag2$$
$$\begin{aligned}
18490&= 10\times43^2\\
18491&= 11\times41^2\\
18496&= 16\times34^2\\
\end{aligned}\tag3$$
$$\begin{aligned}
55223&= 23\times49^2\\
55225&= 25\times47^2\\
55233&= 17\times57^2\\
\end{aligned}\tag4$$
And that's just the results by using certain assumptions, such as two of the numbers $x_i$ being squared are "twin numbers", i.e $47,49$, which differ by $2$. More generous assumptions and relaxing $x_1-x_2 = 2$ would probably net more results.
A: $$2\times2\times\overbrace{41\times41} = \overbrace{82\times82} \approx 7\times\overbrace{31\times31}.$$
This is due the fact that $$\cfrac{82}{31}=2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1}}}}}}$$
is a convergent of the simple continued fraction of $\sqrt{7}$. 
Those two can be combined to get the result. 
