[FOM] Some questions regarding irrational numbers

[Daniel Méhkeri]

> 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. 
[...]
> 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 ?

How about the sum of 2^-n for all even n which are not the sum of two primes.

Doesn't get much simpler! At least in terms of algorithm to compute. And this number is an integer (namely, zero) if and only if Goldbach's conjecture is true (an open question). 

It is also FOM related, since exactly this real number is a favourite "Brouwerian counterexample" to the law of the excluded middle.

Though maybe this was one of the "artificial" examples you had in mind.

Regards
Daniel











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