TITLE: Patterns Are Obvious When You Know To Look For Them... AUTHOR: Eugene Wallingford DATE: June 13, 2016 2:47 PM DESC: ----- BODY:
normalized counts of consecutive 8-digit primes (mod 31)
In Prime After Prime (great title!), Brian Hayes boils down into two sentences the fundamental challenge that faces people doing research:
What I find most surprising about the discovery is that no one noticed these patterns long ago. They are certainly conspicuous enough once you know how to look for them.
It would be so much easier to form hypotheses and run tests if interesting hypotheses were easier to find. Once found, though, we can all see patterns. When they can be computed, we can all write programs to generate them! After reading a paper about the strong correlations among pairs of consecutive prime numbers, Hayes wrote a bunch of programs to visualize the patterns and to see what other patterns he might find. A lot of mathematicians did the same.
Evidently that was a common reaction. Evelyn Lamb, writing in Nature, quotes Soundararajan: "Every single person we've told this ends up writing their own computer program to check it for themselves."
Being able to program means being able to experiment with all kinds of phenomena, even those that seemingly took genius to discover in the first place. Actually, though, Hayes's article gives a tour of the kind of thinking we all can do that can yield new insights. Once he had implemented some basic ideas from the research paper, he let his imagination roam. He tried different moduli. He visualized the data using heat maps. When he noticed some symmetries in his tables, he applied a cyclic shift to the data (which he termed a "twist") to see if some patterns were easier to identify in the new form. Being curious and asking questions like these are one of the ways that researchers manage to stumble upon new patterns that no one has noticed before. Genius may be one way to make great discoveries, but it's not a reliable one for those of us who aren't geniuses. Exploring variations on a theme is a tactic we mortals can use. Some of the heat maps that Hayes generates are quite beautiful. The image above is a heat map of the normalized counts of consecutive eight-digit primes, taken modulo 31. He has more fun making images of his twists and with other kinds of primes. I recommend reading the entire article, for its math, for its art, and as an implicit narration of how a computational scientist approaches a cool result. -----