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.


Take any root of any polynomial "quite long and crazy". You can approximate the root using a sequence of rational numbers and then easily find a series with sum = the root.

Nice example: $$\sqrt 2 = \sum_{k=0}^\infty(-1)^{k+1}\frac{(2k-3)!!}{(2k)!!}$$ (write the Taylor series of $\sqrt{1+x}$)

  • $\begingroup$ Thank you for the example. This is not what I had in mind, as the number was not expressible as a quadratic surd (else it would not have been an open problem whether or not it was algebraic for so long!), but if no-one gives a better example in a couple of days, I'll accept yours. $\endgroup$ – Simon_Peterson Jan 19 '16 at 18:48
  • $\begingroup$ @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. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 19 '16 at 18:55
  • $\begingroup$ Yes, I know, but the point was that the number I am looking for was originally defined as a series, and it was unknown for a long time whether it was algebraic or not. $\endgroup$ – Simon_Peterson Jan 19 '16 at 18:59
  • $\begingroup$ @Simon_Peterson, in any case, my point is that the property "algebraic and sum of a series" is almost trivial. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 19 '16 at 18:59
  • $\begingroup$ @Simon_Peterson, OK, now I see your point. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 19 '16 at 19:01

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