[FOM] Some questions regarding irrational numbers

Rempe, Lasse L.Rempe at liverpool.ac.uk
Fri Feb 27 17:34:42 EST 2009

```> 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.

Yes, I was really looking for something that wasn't merely designed to encode another conjecture. Once we begin forming infinite sums over various sequences, the problem does become a lot easier. However, I do agree this example is particularly nice and simple. It may be that my question on this point wasn't entirely well-conceived.

The decidability of real numbers with exponentiation (assuming Schanuel's conjecture) is presumably relevant here -- it suggests, as one would expect, that it is unlikely that we can 'write down' a number with the properties I suggested, using just basic operations. On the other hand, there are plenty of "natural" functions of a real variable that we could add into the mix, other than exponentiation - should we expect that any of these yield some identities that are true but that we cannot easily verify?

Informally, my question really is: is there a number, conjectured but not known to be an integer, for which the first thought of a high-school student or beginning undergraduate would be "why don't we just compute it?"

I suspect that that is too much to ask. But I would really like to see a number that doesn't obviously encode another conjecture, but rather looks interesting in its own right.

Lasse
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