# Strange Algebraic Number

We call a number algebraic if and only if it is the solution of a polynomial with integer coefficients. A number (complex or real) is transcendental if and only if it is not algebraic.

A while back while reading about transcendental numbers (and the open problem of whether Catalan's constant is transcendental) I recall reading about some strange number, expressed as a series, which actually turned out to be algebraic (with the polynomial being quite long and crazy).

Can anyone give me a reference to it (or perhaps to another similar algebraic number)? I am willing to consider any algebraic number, which was originally defined as a 'nice' series, has no trivial expression as a quadratic surd (or sum of such), and has a long and crazy polynomial, as a satisfactory answer.

Edit: I think this (the proof that the number was algebraic) was a rather new result, i.e. in the past 10 (or at most 20) years.

Nice example: $$\sqrt 2 = \sum_{k=0}^\infty(-1)^{k+1}\frac{(2k-3)!!}{(2k)!!}$$ (write the Taylor series of $\sqrt{1+x}$)
• @Simon_Peterson, you can generalize the example to any $n$-th root. Sum two (or more) roots: $\root{127}\of 2+\root{999}\of 2$ is an algebraic number with a long and crazy minimal polynomial that can be expressed with a nice series. – Martín-Blas Pérez Pinilla Jan 19 '16 at 18:55