Section 44.1 Script
Part 1
\(\textbf{Scene:}\) Seryozha is lying in bed. He has a bandage around his head. He is wounded from battle. He is trying to sleep but cannot. He picks up a crossword but throws it down after a second in frustration. He picks up a letter and writes to his old math teacher.
Seryozha (reading as he writes):
Dear sir, I hope that you are well, and I hope it will please you to know that I am too. I should be clear to say that I am alive, in a way that many of my comrades are not, though I am nearly dead from boredom. The bullet did not kill me, but the time in this bed surely will.
While I think you perhaps do not remember me, I attended your mathematics lectures for a year, and it is with regards to that time that I am writing you now. It was three years ago now, before my service, and you struck me as a knowledgeable and well-learned professor. My friends have suggested to me that I need to occupy myself while I recover, and I can think of no more occupying diversion than the study of mathematics which puzzled me so much in your lecture hall. It is my hope that you can send me some little mathematical pearls to work on while I am laid up.
I am leaving it to your expertise to decide the topic and scope. Whatever may take your interest now, I assure you that it is twice as entertaining as my crossword, and infinitely more meaningful.
Yours,
Seryozha
Skeleton for Khinchin - Part 2
\(\textbf{Scene:}\) Aleksandr Khinchin is in his office, sitting on his large armchair, one knee over another. He begins to read a letter he got earlier in the day and starts to read it as he drinks the last sip of coffee in his mug.
Smiles as he reads the letter, and proceeds to write a letter to his student.
Dear Seroyzha,
Thank you for your letter, I am very grateful and happy that you want to continue learning even while you are at the Front. It is great to see you continuing to pursue your passions. I would be honored to send you a pearl of arithmetic to help you pass the time and learn.
Though I do not know you well, as you were only in my class for a year, and dont know your interests in Mathematics; I hope that you can find interest in the ideas and theorems I am sending to you. The methods you will use are simple arithmetic, but you will find that it will take you some time to understand these problems, as the solutions are not as simple as the methods.
Alas, I am choosing to send you a pearl, which is van der Waerden?s theorem. The idea of this theorem is, if you take a set of all natural numbers, and divide into two subsets, defined by any rule (Even/Odd, Prime/Non-prime, etc.), could you assert an arithmetic progression to these subsets? This problem was first solved by a young student in Gottingen, van der Waerden. Recently, a talented Mathematician of Minsk, M. A. Lukomskaya sent me a much simpler proof of this theorem, which with her permission, I am going to show you.
It has brought me immense pleasure that you have chosen to continue to explore mathematics during your time healing, and I wish you the best, both in the battle and Mathematics.
Yours,
Khinchin
Lukomskaya - Part 3
\(\textbf{Scene:}\) Lukomskaya stands at the front of the room, explaining her newly discovered proof to an audience of other mathematicians.
Dear Dr. Khinchin,
It is a pleasure to hear from you. Of course you may pass my proof along to your student. After all, mathematics is twice the pleasure when shared. Do send along my wishes for a quick recovery as well.
You remember that when van Der Waerden?s theorem was first published, it created quite a stir, for where the best of us had failed to find a solution, a green mathematician had succeeded. Furthermore, while the result was elementary, it was hardly simple. The alternative proof I have outlined below is considerably less complicated.
For posterity, van der Waerden?s theorem is as follows: Let \(k\) and \(l\) be natural numbers. Then there exists a natural number \(n(k,l)\) such that any \(k\)-coloring of the segment of positive integers of length \(n(k,l)\) contains a monochromatic \(l\)-term arithmetic progression.
In essence, we are trying to prove that there exists a function \(f(i) = a+id\) such that \(f(i)\) for all \(0\leq i\lt l\) are of the same colour. We do this using induction on \(l\text{,}\) the length of the progression. The result is trivial for \(l = 1\text{.}\) For \(l = 2\text{,}\) we only need to make sure that there are at least two elements of any colour.
We assume the theorem is true for some \(l\geq 2\) and prove it for \(l+1\ldots\)