[FOM] Some questions regarding irrational numbers

[Timothy Y. Chow]

Lasse Rempe wrote:
> 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?"

Well, then, the Turan hypergraph example might work for you---the only 
problem being that it might be hard to explain the example to a 
high-school student or even an undegraduate.  Let me say precisely what it 
is.  If it has the right flavor, one could hunt more diligently for a more 
elementary example of this type.

I'm following Chung and Graham's presentation in their book "Erdos on 
Graphs."  By an "r-graph" they mean an r-uniform hypergraph, i.e., a 
family of r-element subsets of an n-element set.  Let us define t(n)
to be the largest integer t such that there exists a 3-graph on n vertices 
with t edges which does not contain any complete 3-graph on 5 vertices.  
Then Turan conjectured that

  lim_{n->oo} (n choose 3)/t(n)  =  4.

In principle this is not that hard to explain to an undergraduate, but I 
would predict a lot of glazed-over eyes because it would be so far removed 
from their mathematical experience that the definition would seem totally 
artificial and unmotivated.

A more technical objection is that I'm not sure that anyone has gone so 
far as to predict a rate of convergence, so maybe the limit (which can 
easily be proved to exist) doesn't count as being "computable."

Tim