The idle ramblings of a Jack of some trades, Master of none

Mar 12, 2011


I have learned over the years that what is amusing to one person is far from amusing to another. This is not surprising. Many people - some as young as five years old (see yesterday's post) - have told me that what I find interesting is less than crudworthy to them. Neither trivia nor humour really translates well.

And I do not even mean translating from one language to another. In a recent paper on an algebraic formula for the computation of p(n), the number of partitions of an integer n (a partition of n being any non-increasing sequence of positive integers that add up to n) - by all accounts a major development in number theory, having been an open problem for nearly 80 years - the authors write "We give an amusing proof of the fact that p(1) = 1."

When I saw the proof, I laughed. Hollowly. Here it is (following the statement of their main theorem, for which, see original paper):
In this case, we have that 24n - 1 = 23, and we use the G0(6)-representatives

The corresponding CM points are

Using the explicit Fourier expansion of P(z), we find that

Using these numerics, we can prove that

We have that Tr(1) = 23, confirming that p(1) = Tr(1)/23 = 1.
Quod bloody erat freakin' demonstrandum.


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