Learning Number Theory — like Paul Halmos did

I find that many of my very young “kids” are fascinated to number theory as soon as I start teaching it to them in a ‘friendly’ way, or by way of  “numerical experiments” and then getting them hooked to solving classics in number theory for RMO (Regional mathematics olympiad) and INMO (Indian National Mathematics Olympiad). There is a rich variety of backgrounds of these students here in India.

But, if you (my reader/or my current student/ or even my past student) would like to pick it up in a more serious manner, I reproduce below the famous mathematician, Paul Halmos’ experience as a graduate student (Ref; I want to be a mathematician, An Automathography by Paul Halmos) (his advice is, of course, useful to students pursuing their undergraduate in math, or computer science or also, or those preparing for competitive exams like AMC, AIME, etc.) or even graduate students in such disciplines): Here it is:

…R. D. Carmichael was one of the outstanding members Illinois department. He told me that for a period of a several years there were only three mathematicians on earth who published more than 100 pages a year: G. D. Birkhoff, N. E. Norfund, and himself. His lectures were supremely organized, clearly delivered, inspiring. When I learned from him that 2^{2^{5}}+1 \equiv \mod 641, I rushed home and entered that in my diary. He wrote several books and kept publishing a lot (mainly differential equations), I admired him and wanted to be like him in as many ways as I could. His handwritten p, for instance, was idiosyncratic (the vertical stroke extended almost as far above the level of the loop as below), and I adopted it; to  this day, my p’s look odd.

I fell in love with number theory in Carmichael’s course. He made us prepare a table whose row headings were the first 400 positive integers, and whose column headings, about 25 of them, were items like factorization, sums of squares, sums of primes. Our instructions were to fill in the table, and then proceed to guess (and if possible, prove) as many theorems as we could.

My first research was inspired by Carmichael. He told us about a peculiar question (inspired by perhaps by the four square theorem): for which positive integers a, b, c, d is it true that every positive integer is representable by the form ax^{2}+by^{2}+cz^{2}+dt^{2}(Representable means via integral values of the variables x, y, z, t.) The answer is that there are exactly 54 such forms, and Srinivas Ramanujan determined them all. My question was: which forms of the same kind represent every positive integer with exactly one exception ? I found 88 candidates, proved that there could be no others, and proved that 86 of them actually worked. (Example: x^{2}+y^{2}+2z^{2}+29t^{2} fails to represent 14 only.) The two I could not decide were later settled by Gordon Pall: both x^{2}+2y^{2}+7z^{2}+11t^{2} and x^{2}+2y^{2}+7z^{2}+13t^{2} fail to represent 5 only. No inspiration was needed for this, my first published research, only patience and diligent application of  the techniques Carmichael taught me, but the work gave me a feeling of accomplishment and (badly needed) the confidence that I could do research. I was very proud. I bought 200 reprints; it took me years to find enough people to give them to.

\vdots

By the way, an immortal mathematican, Carl Friedrich Gauss used to say (something like this): Mathematics is the queen of all sciences, and number theory it’s best field. Gauss held number theory in high esteem.

🙂 🙂 🙂

-Nalin Pithwa.

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