[MUSIC] Okay. Now we're going to discuss the proof of Euler's Pythagorean theorem. We have this expression for the denominator, Q(q), which is (1- q) (1- q squared) and (1- q to the 3rd) times etc. If we replace all minuses by pluses, we get the expression for the number of partitions of n into this ten summons. Namely, we have Pd(q) = (1 + q) (1 + q squared) (1 + q to the 3rd) times etc. So, if we Expand this expression, we'll have the same summons as in this expression but taken with signs. Same summons as in the expression for Pd(q), but taken with signs. Okay, and what are these signs? So if we take an even number of factors which are not equal to one in this expression. Then we have a plus one in front of the corresponding summon. And if we take an odd number of factors different from the identity, we get a negative sign. So plus one corresponds to partitions with an even number of rows, and minus ones correspond to partitions with an odd number of rows. Plus ones and negative ones correspond to partitions With an even and odd number of rows respectively. Okay. So the coefficient in front of q to the power n, in this expression, equals to the number of partitions consisting of n boxes with an even number of rows. Minus the number of partitions with n boxes consisting of an odd number of rows. So the coefficient in front of q to the power n in the expansion of Q(q) equals the following thing, the number of diagrams. With n boxes. And an even number of rows. Minus the number of diagrams with n boxes and an odd number of rows. So we need to show that this number is either one or negative one if n is of the form n times 3n minus 1 over 2, or n times 3n plus 1 over 2. And zero otherwise. So if n has the form, k(3k-1). That is, it is a triangular, pentagonal number. Or n = k(3k+1) / 2. And 0 otherwise. To show this, We will construct bijection if this number is zero. Bijection between diagrams with n boxes with an even number of rows, and diagrams with n boxes with an odd number of rows. So if we construct a bijection between these sets, we'll show that this difference is indeed, zero. And if n has such a form, we'll show that the cardinalities of these two sets differ by one. [MUSIC]