2021 Math 3600 Review

229 days ago by calkin

Reviewing the ideas of Experimental Mathematics.  

Many of these ideas are discussed in a series of books by Jon Borwein and others.

Mathematics by Experiment, David Bailey and Jonathon Borwein

Experimental Mathematics in Action, Bailey, Borwein, et al

The Computer as Crucible, Borwein and Devlin

Various other articles over the years by Borwein and others.

Why should we view math as an experimental science?

Because we may be able to discover things that we hadn't thought about before.

Example: We saw striations in the number of Goldbach representations.  Lucy's conjecture was that these could be explained by the prime factors dividing $n$.  Experiments suggested a multiplicative correction factor.  Then some heuristic arguments suggested the Hardy Littlewood correction term $HL(n)$.

Note: we haven't proved that Goldbach is true.   We started with an experiment, to compute the number of Goldbach representations.  We *looked* at the data: in an effective way, by plotting it.

(Aside: there is an excellent book by Edward R. Tufte, "The Visual Display of Quantitative Information" on how to visualize data effectively).

The act of looking at the data, interacting with it, let to an observation about the data, namely that there was structure to the picture, there were striations in the data.  We also saw two horizontal stripes, at 0 and at 2, representing odd numbers that were not/were the larger of a twin prime pair.  We removed these odd number entries from the data, and looked only at the even numbers (which were, of course, the original focus of the Goldbach Conjecture).  We had to find words to describe what we saw.

Once we had identified the strange structure, we could then investigate what might cause it.  In principle, there might be many ways to go about this.  What we did in practice, was to identify the "champions", and see what we could observe about them.  What we didn't do, and might have chosen to do, is to identify the runts, the values of $n$ for which there were few representations, and look at them.  What would we have found?  We suspect, but haven't done the experiment so we can't know whether our expectations match reality, that the runts will be products of 2 and a big prime, or a product of a few big primes.  We haven't considered at all the number of powers of 2 that divide $n$.  We also haven't considered higher powers of 3, 5, 7 etc either.  What experiments could we suggest for a future class to consider regarding prime powers?  Perhaps color the data by which power of 2 divides $n$?  Or which power of 3?

Having identified the fact that divisibility by small primes appeared highly correlated with the striations, we attempted to quantify the effect.  Eyeballing the data for remainders modulo 3, it appeared that we roughly doubled the number of representations of $n$ when $n$ was 0 (mod 3).

This led us to consider how $n$ can be a sum of two primes when it is, respectively, $\equiv 0$, $\not\equiv 0$ (mod 3), which enabled us to come up with a heuristic for why the correction factor for divisibility by 3 should be 2.  The heuristic is actually a proof (of an appropriately stated result) that the average number of representations of numbers divisible by 3 should be twice the average number of representations of numbers not divisible by 3.  (We used a big theorem of Dirichlet, that there are about as many primes congruent to 1 and there are congruent to 2 (mod 3), and looked at how many sums there are in $S_1+S_1$, $S_1+S_2$, $S_2+S_2$, carefully accounting for everything).

This led us to consider similar ideas for other prime divisors, leading to the Hardy-Littlewood correction factor.

After dividing by $HL(n)$, the data looked a lot more coherent.  Perhaps consistent would also be a good word?