0.246810121416...: Is it a algebraic number?

Solution 1:

No, this number is transcendental. The proof by Mahler mentioned in a comment shows this.

A good reference to learn about basic transcendental number theory is the book "Making transcendence transparent: an intuitive approach to classical transcendental number theory", by Edward Burger and Robert Tubbs, Springer-Verlag (2004).

In chapter 1 of the book the proof of the transcendence of Mahler's constant $0.1234\dots$ is discussed. The idea is to show that the "obvious" rational approximations actually are very very close, to the point that they would contradict easy estimates (due to Liouville) for how quickly rational numbers can approximate irrational algebraic numbers. The Wikipedia entry on Liouville numbers discusses Liouville's approximation theorem and related results:

If $\alpha$ is algebraic of degree $d\ge 2$ then there is a constant $C$ such that for any rational $p/q$ with $q>0$, we have $$ \left|\alpha-\frac pq\right|>\frac{C}{q^d}. $$

Actually, there is a bit of work needed here. The estimates the book discusses together with a strengthening of Liouville's theorem give the proof for Mahler's constant, and the same argument works for the number you are asking.

The strengthening we need is due to Klaus Roth in 1955, and he was awarded the Fields medal in 1958 for this result.