This is a good question with lots of paths to follow.

There is a connection between Fermat's Last Theorem and the differential ring realised in an effective potential.    An effective potential can be expressed as a section of a finer bundle or section of a line on a manifold.    This is explained in detail in

page 54:

T. Ohsaku, Dynamical mass generations and collective excitations…, arXiv, 2010:

http://inspirehep.net/record/779121/files/arXiv%3A0802.1286.pdf

With the section of a line on a manifold, there are many physical applications to consider, e.g., section of a line in manifold learning, where one looks for patterns in digital images (a good place to look for such patterns is in sections of lines in images mapped to corresponding manifolds).

As an aside, the principle of least action in classical physics can be traced back to Fermat's theory of optics (1662).    See

J. Rosenberg, Dualities in field theories and the role of K-theory, University of Maryland, 2011:

http://www.math.umd.edu/~jmr/Kyotolecs.pdf

For a very detailed, extensive exposition of the role of mathematics in physical science, see G. Boniolo, P. Budinich, M. Trobok, Springer, 2005:

http://www.jep.ir/fa/files/downloads/mathetics_in_physic.pdf

See, for example, page 14.

Recent experiments have determined that the universe is not just locally euclidean, but is actually euclidean in general. This surprised me. For many years the best guess was that the universe was hyperbolic.

By Wilson's Theorem do you mean n prime if and only if  n divides (n-1)! + 1? In fact this is only a way to say that in every finite abelian group, the product of all elements is equal with the product of elements of order 2, applied for the multiplicative group of the field Z/nZ, for n prime. (The reciprocal is easier, because in non-prime Z/nZ there are zero-divisors...) If this is the result you mean, I don't know what physical interpretation this could have - because finite cyclic rings are not really so often meat in physics. Finite fields are possibly meat in the world of Quarks - or if not since now - maybe there will be present there some day.

Fermat's Little Theorem p is prime iff p divides ap - a seems to stands more natural together with Wilson's Theorem, because is quite similar. It means only that an element in a finite Group has an order that divides the order in the group. This possibly appears also in physics quite often.

As about Fermats Last Theorem, I can give only one association: the curve xn + yn= 1 has a polynomial parametrisation only for n = 2. One can get the pythagorean triples

(m-n)2 ; 2mn ; (m+n)2 only by such a parametrisation. This could have some implication in physics. The source

http://en.wikipedia.org/wiki/Pythagorean_triple

presents some connection with spinors.

@ James

Thank you James for the good sources, specially for the last one.

Dear James, 

The links you have put particularly as dear Mihai said, the one that deals with the wider interplay between mathematics and the physical sciences was a good read and interesting manuscript. 

Dear @Mihai and @Dejenie,

Many thanks for your comments.   Frankly, I was quite surprised at the range of connections between Fermat's Last Theorem and the physical sciences.    It is a beautiful subject to pursue and it is nice to see some powerful outcomes of Fermat's mathematics.

@  Lawrence Margulies

This is very interesting what you write. If there were evidence for a Universe which is globally euclidean, this matches an older thought of mine, that I always communicated on RG. I think that the Big Bang is only a local phenomenon. It might be the start of the Universe we could see so far, but not for the whole Universe.

Can you give us some references for the new evidences for a globally euclidean universe?

Here is a good link to the actual data (microwave background radiation) which has led to this new result.

http://map.gsfc.nasa.gov/

@Mihai

Yes. I also agree with your idea about the big bang being a local phenomenon. This would solve the entropy problem of an infinite universe. A big bang happens in a locally low area of entropy in the universe. There are a number of theories that consider this, however it will probably be another 10 years or so before there is experimental data to test these theories.

Thank you Lawrence. I have thought before only at density - but you are right, it might be the entropy. It would make more sense, also from the termodynamic point of view. In particular, almost no matter means also zero entropy.

> Could Fermat and Wilson's Theorem be the reason space is locally Euclidiean, and if so, does curved-spacetime disobey their theorem?

According to available observational data we do live in "curved-spacetime", so in fact it does not "disobey" these theorems. Moreover, from the point of view of classical physics relation of these number theory theorems to some specific geometry of some Universe (in which we happen to have this discussion) is not obvious/very plausible; finally, curved or not, the spacetime is at least considered a manifold at this level, hence by definition it is locally Euclidean.

At a deeper level -- it's probable that at small enough (Planckian) distances the Universe stops looking like a manifold. Hence it is mathematically interesting to ask what universes are in principle possible -- whether any could exist that are non-geometrical on all scales. And for questions like this it may turn out that some theorems (e.g. Fermat's or Wilson's) could prohibit such a possibility (like anomaly cancellation imposes restrictions on the number of quark and lepton families in Standard model or on geometry/dimensionality of target space in String theory). This is however too distant from our current understanding, not to mention from experimental verification. And regardless, this is immaterial for the Universe we know, as we know that it does look like a (pseudo-)Riemannian manifold on scales above atomic.

I don't think I understand your question, but clearly curved space time can be mathematically formulated, and so does not contradict a theorem.