# How to appreciate Fermat's last theorem?

I am someone who is not a Maths major, these days (during the summer) I am attracted to Fermat's Last Theorem. I understand that there is no whole number solution to the equation $x^n + y^n = z^n$ for $n\gt 2$ but doing a simple calculation we can have a non-whole number solution?

So, my question is this: What is that the whole numbers are that important?

• Best appreciated with a glass of chilled pinot grigio... – copper.hat May 21 '15 at 2:24
• As far as we can tell at the moment, the truth of FLT relies on some fairly deep mathematics. There may turn out to be more elementary basis for it, but it seems somewhat unlikely, given that people have been looking for that for centuries, and haven't found it. So, failing that development, @copper.hat's suggestion of a glass of wine seems as good as any. – Brian Tung May 21 '15 at 2:26
• As for why whole numbers are important, among other things, they are the building blocks of the rest of the numbers. – graydad May 21 '15 at 2:28
• What is your definition of "important." It's a vague term. Mathematicians have been interested in integer solutions to $x^2+y^2=z^2$ since at least Pythagorus. Is that because it is "important?" I don't know, but there is some fascinating mathematics that comes from looking for integer solutions to that equation. The same happened with Fermat. People were interested in the question, and they developed a lot of really interesting and deep mathematics trying to solve it. – Thomas Andrews May 21 '15 at 2:35
• What is that the whole numbers are that important ? - Because finding a real solution to Fermat's equation is trivial. – Lucian May 21 '15 at 5:17

Clearly there are real and algebraic solutions. Just pick any value for $x$ and $y$ you want, and then solve for $z$ (possibly in the complex numbers). Since integers and rationals were the "first numbers" and Diophantine equations perhaps the "first equations" it makes some sense that we search for rational and integer solutions to Diophantine equations. In this case, because of homogeneity (each term in the equation has the same degree), rational solutions imply integer solutions (just clear denominators to get integer solutions from rational ones as so:

$$\left(\frac{a}{b}\right)^n+\left(\frac{c}{d}\right)^n=\left(\frac{e}{f}\right)^n \iff (adf)^n+(cbf)^n=(ebd)^n.$$

And the existence of integer solutions implies the existence of whole number solutions. First, if the exponent $n$ is even, then any solution $(x,y,z)$ means there is a solution $(|x|,|y|,|z|)$. Second, if $n$ is odd and for instance $(-x)^n+y^n=z^n$ then $y^n=x^n+z^n$ hence $(x,z,y)$ is a whole number solution, or else if $(-x)^n+(-y)^n=x^n$ then $x^n+y^n+z^n=0$ which isn't possible for nonzero $(x,y,z)$ so that case can't occur anyway. Any integer solution $(x,y,z)$ will yield a whole number solution $(a,b,c)$ by negating or permuting the $x,y,z$ as appropriate (try it out!).

Among Diophantine equations, $x^n+y^n=z^n$ is at once both very simple in form and surprisingly deep in its truth (if the length and complexity of its proof are any indication), which sets it apart from other Diophantine equations. Personally, I'm not terribly interested in all of these Diophantine equations, and in fact I find the Modularity theorem (which was a critical component in the proof of Fermat's last theorem, and previously called the Taniyama-Shimura-Weil conjecture) more interesting. But this illustrates an important point that is touched on again and again: the search for a proof of Fermat's last theorem is what inspired the development of a lot of modern algebraic number theory, and so the theorem is very significant in terms of inspiration and history.

• That FLT "inspired the development of much of modern algebraic number theory" is a great exaggeration, the byproduct of too many romantized historical fairy tales. – Bill Dubuque May 21 '15 at 5:45
• @bill nevertheless, it's a perfect example of the true importance of hard problem proofs: new techniques are developed, which are more important on their own than the theorem for which they were invented – mattecapu May 21 '15 at 16:12

Integer solutions of a problem are essential in many areas. You wouldn't like that the number of persons to fit in the room be $13.5$, would you ?

But besides this obvious pragmatism, it turns out that all research around integer solutions of equations, the so-called Number theory, ends-up in numerous beautiful (and difficult) extensions that make the delight of mathematicians.

Fermat's last theorem is a perfect example, having required an army of the finest minds for several centuries before the answer was found.

Interestingly, Number theory makes a heavy use of techniques drawn from the theory of real and complex variables.

Do you know the website Wolfram Alpha? You can use it to play around with equations, which will perhaps deepen your appreciation of Fermat's so-called last theorem (both words are problematic, but that's a different can of worms).

Part of the reason whole numbers are important is because the problem becomes trivially simple if the requirement for integers is dropped. Consider for example $n = 3$. Go to Wolfram Alpha and type in solve x^3 + (5/2)^3 = 7^3. One of the three solutions is $$\frac{3 \root 3 \of {97}}{2}.$$ Now change (5/2) to (46/7). Play around with it some more. You'll become convinced that, without the requirement of integers, there's always a non-trivial solution.

Then try this: solve x^3 + y^3 = 7^3 in integers. There are solutions, but these require $x$ or $y$ to be zero, which is no fun, right?