Abstract:

       In 1760, Euler noticed that if you multiply the
       polynomials 1-x, 1-x^2, 1-x^3, 1-x^4, etc., then the
       result converges to 1-x-x^2+x^5+x^7-x^12-x^15+x^22+x^26-... 
       (can you guess the formula? it involves the 
       "pentagonal numbers" (3n^2+n)/2). This formula has 
       some unexpected applications, for example, 
       it provides an easy way to calculate 
       the number of partitions of a given positive
       integer into (unordered) sums of positive integers.
       (For example, 5 has 7 such partitions: 5, 4+1, 3+2,
       3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; 10 has 42 such
       partitions; but how many partitions are there for 20?
       for 50? for 100? I will show how to solve these problems
       by a two-minute computation; well, you may need three
       minutes for 100.)

       Some fifty years after Euler's discovery Gauss and
       Jacobi notices that if you multiply the cubes of the
       same polynomials (1-x, 1-x^2, etc.), you will get
       something still more remarkable that Euler's identity.
       (Take a pencil and a piece of paper and try to calculate
       and to guess the answer.) This looks especially
       unexpected, if you notice that the product of squares of
       our factors does not seem to provide anything
       interesting. (In 1972, a well known physicist Dyson
       compiled a list of exponents of the Euler product for
       which there was a known (to him) formula: 1, 3, 6, 8,
       10, 14, 15, 21, 24, 26; we will discuss this briefly).

       I spoke about this a couple of years ago, but now I want
       to show to you a very simple and very elegant proof of
       the Gauss-Jacobi identity found in 1984 by Zinovi
       Leibenzon. You will like it.

       This material is elementary, no special knowledge is
       required. Well, if you know how to multiply polynomials,
       it may help.