[FOM] Some questions regarding irrational numbers

[Timothy Y. Chow]

Lasse Rempe wrote:
> 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?

Here's one I just thought of: the exponent of matrix multiplication.  
That is, what is the least number d such that the product of two
nxn matrices can be computed using O(n^d) arithmetic operations?
It is widely conjectured that d = 2.

By the way, perhaps a suitable definition of "artificial example" here 
would be something like this.  For an example to avoid the charge of 
artificiality, the standard way of stating the conjecture has to have the 
form "x is an integer" or "x = 17" or something like that.  Thus the 
Goldbach conjecture is an artificial example because nobody ever states 
the conjecture in such a form unless they are told deliberately to 
manipulate the conjecture into that form.

The exponent of matrix multiplication is natural in this sense.  However, 
there is no known "reasonable" algorithm to compute its value to any 
desired accuracy.  Thus it may not have the right flavor of what you are 
hoping for.

There are other examples from extremal combinatorics.  For example, 
there's a problem of Turan on hypergraphs for which Erdos offered $1000;
small special cases have the form "such-and-such a limit is 4" where we 
have no proof that the limit should be any kind of nice real number, let 
alone an integer.  These are probably too "complicated" for your purposes 
though.

Tim