| > > ...Fermat's last theorem has been shown to depend on a "structural
> > conjecture" in number theory. ...
>
> However, the author made no explanation of the "conjecture".
>
Which implies that either he didn't understand it either, or that it was
too complicated to put into the article he was writing :-).
> I virtually don't know anything about number theory, but I'm interested
> in knowing what the author was talking about. Could anybody tell
> me what the author was refering to?
>
Fermat's Last Theorem is easy to state ("the equation x^n+y^n=z^n has no
solution in integers x,y,z,n for n>2") but incredibly difficult to deal
with. Number theorists have been banging away at it for many years. In the
process they have invented many powerful tools, and discovered many
powerful concepts, for dealing with infinite classes of numbers with
certain properties.
Some of the concepts have to do with 'structure'. As an example of a
structural concept, many integers are 'square-free' in that they have no
factor that appears twice in its factorization (10 = 2*5 is square-free; 12
= 3*2*2 is not).
The theorists observe many *conjectures* that they cannot yet prove. So
what the writer is saying is that Fermat's Last Theorem has been
demonstrated to be logically equivalent to another problem that they can't
solve yet either, but which perhaps is easier to attack because it only
deals with a certain class of numbers with a common structural property,
instead of ALL numbers.
As to what specific conjecture the writer was referring to, I don't know.
|