Monday, June 6, 2016

Brian Hayes in Pursuit of Prime Numbers

Ulam Spiral via WikimediaCommons

There are many questions, notably in theology, but in science as well, that may NEVER give way to human reason; their formulation is so far beyond the limitations of measly, squishy brains and hard-wired computers. One of the beauties of math, however, is that such a high percentage of its questions ARE amenable to comprehension with mere human logic and persistence.
Just maybe understanding prime numbers is one such subject (...though maybe it is not!).

Brian Hayes' latest post on the non-randomness of primes is a beautiful read (pretty typical for Brian actually). I can't pretend to comprehend 70% of it :-(  but that doesn't prevent me from appreciating and sensing the work he has put in to it and the direction it takes.
With its visual power (reminiscent of the Ulam Spiral above), Brian's post yields, even without a full understanding, that ineffable sense that SOMETHING significant is going on here... something tantalizingly, almost tauntingly just within, or, just beyond human grasp? And with a little more time, or effort, or computer power, perhaps we can tap into it. The primes toy with us, tease us, and Brian falls under their siren spell.

Hayes writes that he's been working for a couple of months to get to the point of what he presents in the post, a continuation of previously-discussed recent findings about non-randomness in the order of primes. There is of course the far-more famous case of Andrew Wiles wiling away secretly for 6+ years to prove Fermat's Last Theorem -- I admire the dogged, focused persistence and willingness of humans to secrete themselves away with their own brains as lone company to wrestle with such abstract knowledge, not even knowing if anything useful may result from it... the passion for knowledge/pure-math for its own sake.
What does it mean that primes, the building blocks of our number system, seem to have order/pattern, even if we can barely discern it; and yet any such order/pattern seems to change/evolve as one goes farther and farther out in the run of primes toward infinity? Maybe by now Erdös has devoured 'God's Book' and knows all these answers, but we're still scratching our heads in confused wonder.

When the original Lemke Oliver/Soundararajan work was reported to much fanfare, I wrote that it looked like the sort of thing that would swing open the door (floodgates?) to much further study. Brian's work is likely just one of the many paths one might go down. It is the sort of thing even amateurs, with some computer skills and interest, can play with almost endlessly... and, just maybe, strike gold.

Do prime numbers exist only in our heads, amenable to full self-discovery, or do they lie in some more mystical Platonic realm forever just beyond our reach? I wish I knew. In the end, both Brian's hope and frustration is palpable:
"The complexity of the mathematical treatment leaves me feeling frustrated, but it's hardly unusual for an easily stated problem to require a deep and difficult solution. I hang onto the hope that some of the technicalities will be brushed aside and the main ideas will emerge more clearly with further work. In the meantime, it's still possible to explore a fascinating and long-hidden corner of number theory with the simplest of computational tools and a bit of graphics."
Anyway, read what all Brian has done. It's 23 pages (if printed out) of deliciousness, even though the last 1/3 of it may be especially tough going for general readers:

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