Introduction

Let Ext denote the E2 term of the classical Adams spectral sequence for the stable sphere mod 2 [Al]. Then Ext is a bi-graded algebra over the field of 2 elements. Write e(s, t) for the rank of this vector space in bi-grading (s, t).

It would be very helpful if we could find a method to compute these ranks, with sub stantially less work than it now requires to obtain Ext itself, or Ext with much of its algebra structure. Perhaps this is not a well-posed problem; we have certainly not “solved” it. What we do have to present at this time is some interesting observations about these numbers.

First, the numbers e(s, t) are not randomly distributed in the plane (or in any part of the plane). We call this “Poisson's theorem”.

Next, the alternating sums of the e(s, t) can easily be computed, in the form of a simple generating function. We call this “Euler's Theorem”.

If these alternating sums are defined in the right way, they are all O's and l's, and the gaps between the l's can be described in terms of the Bernoulli numbers. This gives us “Bernoulli's theorem”.

Topologists will already have noticed that our theorems are not really about the Adams spectral sequence, but rather about its E2 term.

Our title may seem facetious, but it signals to the reader that our results have a classical (and elementary) flavor, and that this note is meant to be enjoyed.