google.co.nz/logos/2011/pierre ...
Alt text is: I have discovered a truly marvelous proof of this theorem, which this doodle is too small to contain. 
Triggers google.com/#q=Pierre+de+Fermat ...
on Google . com today. 
This documentary will explain why the doodle is too small to prove Fermat's Last Theorem.
topdocumentaryfilms.com/fermat ...

The formula is correct because X^N+Y^N does not = Z^N where N>2 but it is also true that X^N+Y^N does not = Z^N where N<1
with the proviso always that X Y and Z are all integers.

I agree with Edward R since N>2 and it can also be N<2 
Yeah I don't know why people think it would work with n=1. The formula is where X and Y are legs and Z is the hypotenuse of a triangle. So the example 3^1 + 2^1 = 5^1 is obviously wrong because the legs of a triangle can't equal the hypotenuse. 
Fermat's Last Theorem really has nothing to do with the legs and hypotenuse of a right triangle. Certainly, in the special case where N=2, then it becomes the Pythagorean Theorem, and the equation holds true for all integer values of X, Y and Z. However, when N is anything but 2, the equation is an inequality, and is more of a mathematical abstraction, and is not directly applicable to 'real' things, such as triangles. 
Pythagorean theorem states that X^2+Y^2=Z^2. That is all that it states. This fact does not necessarily mean that powers other than 2 would not also be equal.
It would require a little more proof than simply comparing this to the pythagorean theorem to determine if the statement is valid or not. 
Fermat's Theorem and mathematics are a side plot in "The Girl Who Kicked the Hornet's Nest" by Stieg Larsson. Lisbeth Salander is trying to work out the proof herself without looking it up. 
Some examples that deserve consideration...
12^1 + 17^1 = 29^1 12^7 + 0^7 = 12^7 
Sorry, John, those examples do NOT deserve consideration. Fermat's Last Theorem specifically states that no three positive integers x, y, and z can satisfy the equation x^n + y^n = z^n for any integer value of n > 2. In your first example n=1, so it doesn't count, and in your second example y=0, so it doesn't count. 
I agree, Barry, but some people are saying simple for cases of n=/=2, and the Doodle fails to specify positive integers. 
does anyone know anyting about a more general case: (the minimum number of monomials in each case?)
for example:
x^5 + y^5 + z^5 + .... = n^5

the real thing now is to find the minimum number of such monomials. 
Santos, that would be an extension of Fermat's Last Theorum, which has only recently been proven. So, no, I don't think anyone does. 
It is true that the doodle does not specify positive integers. But many of the posts regarding the Pythagorean theorem do not specify that the statement is only true if x, y, and z are sides of a right triangle, with z being the hypotenuse. If z is the longest side of a triangle, and x^2 + y^2 < z^2, then the triangle is obtuse; and if z is the longest side, and x^2 + y^2 > z^2, then the triangle is acute. These are all based on angles of the triangle, particularly the angle measure opposite the longest side.
"Pythagorean relationships" are found in many areas of mathematics.

It's not in the directory google.com/logos/
BTW.. 