[FOM] Some questions regarding irrational numbers

Rempe, Lasse L.Rempe at liverpool.ac.uk
Tue Feb 24 17:05:31 EST 2009

While working on an informal introduction to algorithms for an upcoming book, I became curious (for didactic reasons) about a circle of questions that I describe below. I was hoping that FOM readers may find them interesting and might be able to share some further thoughts / answers / references.

It is a well-known open problem whether the number pi + e is irrational. In other words, it is unknown whether any term of the sequence n*(pi + e), for integer n, is itself an integer. My, somewhat vague, question is: what are the "simplest" sequences that can be defined by a closed-form expression and for which it is not known whether any of the terms are integers? 

The above is not a bad example, but it does contain two well-known transcendental constants. It might be nice to avoid this. For example, I am not aware of any results excluding the possibility that ln(ln(n)) is an integer for some integer n; but then again I am not sure where I would look for such a theorem. What other sequences come to mind? And, separately, what is a good definition of "simple" here?

A related question, which may be closer to matters of FOM, is what is the "simplest" number that is conjectured to be an integer, but not known to be so? It seems unlikely that we can "explicitly" write down such a number; on the other hand, we can create some more or less "artificial" numbers with this property. 

There are also some constants e.g. in analysis that are conjectured to have some explicit integer value. Usually these constants are not very explicit both in their definition and in that there is no easy way to approximate them.

So, to make the question a bit more precise: what are some "simple" and "natural" examples of definitions of a real number x, such that:
a) there is an algorithm to approximate x up to any degree of accuracy;
b) it is an open question whether x is an integer or not ?


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