[MaeveChondrally (OP)]

No offense to andrew Wiles and his very interesting but incomprehensible proof. and all those pure mathematicians who demand an exact solution when an infinitessimally close approximation will sufffice.

a^n+b^n=c^n
for any integers a,b,c and any n integer greater than 2.  the theorem states that there are no solutions.
wiles proves this in his magnum opus and indeed there are no exact solutions.
look
2^5+3^5=3.07515165743^5
if we admit real solutions
then we can say that for some integer k,   (10^k*a)^n+(10^k*b)^n=(10^k*c)^n where 10^k*c is truncated to k+1 digits and it becomes an integer
so we could write
2000000000^5+3000000000^5=3075151657^5
is accurate to 1 part in 10^9 which is accurate enough for most scientific calculations and engineering calculations
so if we are pragmatic,  there is an infinite number of infinitessimally close solutions to Fermats last theorem for every n.  k should be greater than about 6 to achieve an accuracy of 1 part per milliion.

best wishes and happy new year
Richard Belshaw
aka
Maeve Chondrally

[evan_au]

What made Fermat's last theorem so hard is that it demands "a solution in integers".
- This demanded breakthroughs in linking previous separate areas of mathematics

If they merely wanted solutions on a 10-digit calculator, all it needs is a 10-digit calculator (and a lot of patience).
- Modern computers usually use 64-bit floating-point numbers, which are accurate enough to prove that the 10-digit calculator is wrong

[charles1948]

Is there a good reason why Mathematicians seem overly concerned with "integers" -  ie, "whole numbers"

And as a result of their concern, have spent over three centuries trying to prove, or disprove, Fermat's Last Theorem.

As the OP points out, sufficiently good, or sometimes even exact solutions, can be obtained if Fermat's Theorem is allowed to admit the existence of fractional numbers. 

So why this demand for a "whole" number solution.  The Universe isn't made up of "whole numbers"!

This shown by a simple fact:: Universal entities as Spheres and Circles, don't have a "whole number" relationship between their diameters and circumferences.  The relationship involves the infamous "Pi".

And "Pi" is not a whole number. Heck, its not even any whole fraction, but a kind of indeterminate value. Which can't be pinned down by maths.

And as far as I know, mathematicians aren't getting stressed out by Pi  So why have they spent three centuries
beating their heads out over Fermat?

[Bored chemist]

there is an infinite number of infinitessimally close solutions to Fermats last theorem

What is the use of an infinite set of slightly wrong answers?

Which can't be pinned down by maths.

I think you will find that maths has it pinned down in a number of ways.
If you were not aware of that, why did you feel qualified to comment on it?

And as far as I know, mathematicians aren't getting stressed out by Pi  So why have they spent three centuries
beating their heads out over Fermat?

It took them until the 1760s to prove that pi isn't rational That's well over 3 millennia of work.
By comparison, solving Fermat's theorem was quite quick.