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The smallest solution for x is given by
Numerator[FromContinuedFraction[ ContinuedFraction[ √ a , (If[EvenQ[#], #, 2 #] &)[ Length[Last[ContinuedFraction[ √ a ]]]]]]]
This is plotted below; complicated variation and some very large values are seen (with a = 61 for example x 1766319049 ).

And if more than two sine functions are involved there no longer seems to be any particular connection to generalized substitution systems or continued fractions.

The continued fraction map x Mod[1/x, 1] discussed on page 914 becomes repetitive whenever its initial condition is a solution to a quadratic equation.

As discussed on page 903 this sequence can be generated by applying substitution rules derived from the continued fraction form of h .

universally been assumed that with the continued development of mathematics any of these questions could in the end be answered.
… And it is my strong suspicion that in fact of all the current unsolved problems seriously studied in number theory a fair fraction will in the end turn out to be questions that cannot ever be answered using the normal axioms of arithmetic.

The n th term in its continued fraction representation turns out to be 2^Fibonacci[n - 2] .

My classical English education—in elementary school (Dragon School) and high school (Eton)—emphasized such pursuits as writing, and exposed me to a certain range of subjects, a remarkable fraction of which have ended up being useful, especially in the historical research for this book. … It has also been a continuing source of further encouragement to see just how broadly and deeply the worldwide Mathematica community has been able to make use of the fundamental ideas that I have embodied in Mathematica.

Any real number x can be represented as a set of integers using for example
Rest[FoldList[Plus, 1, ContinuedFraction[x]]]
but except when x is rational this list is not finite.

In connection with his study of continued fractions Carl Friedrich Gauss noted in 1799 complexity in the behavior of the iterated map x FractionalPart[1/x] .

As discussed on page 944 , whenever a is not a perfect square, there are always an infinite number of solutions given in terms of ContinuedFraction[ √ a ] .