# Transcendental Continued Fractions

Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\ge c/p^n$ for any $c$,$n$. Naturally, one asks whether it is possible to do the same thing using continued fractions rather than decimals. I.e. we should conjecture that [$1!,2!,3!,...$] and [$1,2,4,8,...$] are transcendental since they converge 'very quickly'. However when I actually tried to construct a transcendental number using continued fractions, I could only do so using a very cumbersome recursive definition of the form $$a_{n+1}=n\cdot q_n^n$$ Where $q_n$ is the denominator of the $n^{th}$ partial convergent. This ensures sufficiently fast convergence to use 'Liouville's criterion'. Questions:

1) Are any of the two numbers I listed really transcendental? How is this proved?

2) If not, what is the simplest example of a continued fraction that is transcendental?

3) What are the conditions (necessary or sufficient) on the growth rate of {$a_n$} to ensure transcendence? I am aware that Alan Baker (RIP) has discovered a sufficient condition on $q_n$ that ensures transcendence. What other results have been found as of yet?

• It is not clear what you mean by "the simplest example of a continued fraction that is transcendental", however I guess that $$e=[2;1,2,1,1,4,1,1,6,1,1,8,1,1, 10, …]$$ is simple enough. Mar 31 '18 at 18:47
• How about the simplest example where we conclude a number is transcendental from its continued fraction. That is, without evaluating the continued fraction in terms of previously known transcendentals. Mar 31 '18 at 19:24
• Literally the same argument as in your motivating example (Liouville numbers) works for CF: if the coefficients increase insanely fast, then you have rational approximations that would be too good for an algebraic number. Mar 31 '18 at 21:18
• Maybe the OP is asking: is there an exponential (or sub-exponential) type limit to how fast the CF coefficients can grow for algebraic irrational numbers? Apr 1 '18 at 7:54