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I am fascinated by the fact that "important" irrational numbers like the golden ratio, base of the natural exponent, pi, square roots have a "regular" representation as an infinite continued fraction.

Is there some correspondence between "having a regular ICF representation" and the "importance" of a number, whatever the meaning of "importance" might be? Is this a thing, or is such connection spurious and you could find a "regular" representation for any irrational number if you wanted to?

The opposite question also interests me: if I choose some "regular" representation like [1;1,2,3,2,3,4,3,4,5,4,5,6,...] what are the chances for it's not going to be a square root of some number, for example?

vasily
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    It won't be a quadratic irrational, since those are the numbers with eventually periodic continued fractions. – Ethan Bolker Nov 01 '21 at 00:01
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    Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Nov 01 '21 at 00:01
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    There is no known pattern in the continued fraction for $\pi$. Lagrange proved that the continued fraction representation of a positive real number $x$ is eventually periodic iff $x$ is the root of a quadratic with integer coefficients. There is a big subject called diophantine approximation that grapples with questions like these. – Rob Arthan Nov 01 '21 at 00:09

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