The discussion of Putin’s proof gave me an email from Alexis Tsipras, who just resigned as prime minister of Greece and is busy with the general elections of September 20 soon. Rather than reporting on it, I might as well fully quote it.
Subject: My proof of Fermat’s Last Theorem
Date: Fri, 28 Aug 2015 11:58:03 +0100
Google Unique Message Identifier: 23DFGA@671
Thank you very much for your discussion of President Putin’s proof when he was a youngster of Fermat’s Last Theorem. I know his mother Vera Putina very well. The Putin family has a vacation home here in Greece, and she can stay there on the condition that she immediately leaves when Putin himself comes down. She has shown me his proof too. I can only agree with your conclusion that it shows how smart President Putin was when he was young.
Putin’s proof inspired me to find a proof too. I am sometimes exhausted by the tough negotiations with the European Heads of State and Government, if not with members of my own party. Thus I resort often to a sanatorium for recuperation. Thinking about such issues like Fermat’s Last Theorem helps to clear my mind from mundane thoughts. I was very happy last Spring to indeed find a much shorter and more elegant proof.
For the theorem and notation I refer to your weblog. My proof goes as follows.
Theorem. No positive integers n, a, b and c can satisfy an + bn = cn for n > 2.
Proof. (Alexis Tsipras, April 31 2015)
Let us assume that an + bn = cn holds, and derive a contradiction.
There are two possibilities: (1) n is even, or (2) n is uneven.
(1) If n is even, then we can write A = an/2 and B = bn/2 and C = cn/2 such that A, B and C are still integers. Then we get the following equation:
A2 + B2 = C2
This equation satisfies the condition that n = 2, and thus it doesn’t satisfy the condition n > 2.
(2) If n is uneven, then we can write A = a(n-1)/2 and B = b(n-1)/2 and C = c(n-1)/2 such that A, B and C are still integers. Then we get the following equation:
a A2 + b B2 = c C2
This equation does not satisfy the form of an + bn = cn so that it falls outside of Fermat’s Last Theorem.
In both cases the conditions of the theorem are no longer satisfied. We thus reject the hypothesis that an + bn = cn holds. Q.E.D.
This is much shorter that President Putin’s proof. And, I prove it while he only came close. I have been hesitating to tell him, fearing that he might become jealous, and be no longer willing to support Greece as he does in these difficult times for my country. Now that you have confirmed how wonderful his proof at only age 12 was, I feel more assured. Will you please publish this proof of mine too, like you did with President Putin’s proof ? I have put my best efforts in this proof, just like at the negotiations with the European Heads of State and Government. Thus I hope that it will be equally convincing, if not more.
After the next elections I will probably be exhausted again. I would like to work on another problem then. Do you have any suggestions ?