Today, I learned of a news story that a professor in Russia claims to have solved Fermat's Last Theorem in 3 lines. The story has also been covered in Italy here and here. (For those who don't speak Russian, here is an English translation. For those who don't speak Italian, here and here are English translations of the Italian articles).
It looks like the Russian article is the only one that gives the details of the proof. So, I am relying on this translation.
Let me start by saying that it is very important to be skeptical of anyone claiming to have a solution. In May, the Manilla Times stated that a local man found a mistake in the proof by Andrew Wiles. This of course turned out to be a hoax.
There are two tell-tales sign that the proof is most likely flawed. First, the professor who claims to have found the proof is not an expert in number theory. Second, the proof has not been reviewed by anyone who is an expert of number theory. From the mathematician Gabriel Lame to the very famous Leonhard Euler, it is very easy to make mistakes when it comes to number theory. Even Fermat made a mistake when it came to his conjecture about what are today known as Fermat primes.
Now, onto his proposed proof. After reviewing an English translation of the article, I found (no surprise) that the proof is not a valid demonstration of Fermat's Last Theorem.
First, let me summarize the proof:
(1) Assume that Fermat's Last Theorem is true.
(2) Then, there exists xn + yn = zn with n ≥ 3.
(3) For any x,y, we can create a right triangle and so we can suppose a value r such that:
r2 = x2 + y2.
(4) Since n ≥ 3, we know that z is less than r. [I will add the details for this later]
(5) We can construct a triangle based on z,x,y. Since z is less than r, we know that the angle opposite z (let's call it angle B) must be less than 90 degrees (and of course, greater than 0 degrees).
(6) Now, using the Law of Cosines, we know that:
z2 = x2 + y2 - 2xycosB.
(11) Since B is greater than 0 and less than 90, we know that cosB cannot be a whole number. [See here for a Cosine look up table]
OK, so if the steps are valid, is Fermat's Last Theorem proven? Not at all. Just because cos B is not a whole number doesn't prove that z is not a whole number. That is ultimately what must be proven.
The professor must either prove that cos B is irrational or he must show that based on 2xycosB, z cannot be a whole number.
Since cos B is a continuous function, it necessarily covers rational values and therefore it is quite possible that 2xycosB is a whole number.
Thanks go out to David (see the comments section) for providing his thoughts.