22
If these numbers are arranged in ascending order, a different pattern may be seen in their differences: I
I
2 5 7 12 15 22 26 35 40 51 57 70 77 92 100 3 2 5 3 7 4 9 5 II 6 13 7 15 8
The alternate differences form the natural numbers, 1,2, 3,4, 5, 6, 7 ... and the odd numbers, I, 3, 5, 7, 9, II, 13 ... A very beautiful and important theorem was discovered by Euler, which involves the complete sequence in a surprising way. He started to multiply out the infinite product: (I - x)(1 - x 2 )(1 - x 3 )(1 -
X4) •••
and discovered that the first few terms were: I - x - x2 + X S + x7 -
Xl2 -
XiS
+ ...
At first he felt unable to prove, except by this informal induction, that the indices of the powers of x were indeed the pentagonal numbers, 99