Here's To Amateurs... and Professionals at Play

Astronomy is often considered the best science for amateurs because over the centuries amateur or backyard astronomers have contributed so many important findings to the field. The heavens are so expansive that they offer many niches for even backyard astronomers to make significant discoveries, or be involved in "citizen" science.

Mathematics... not so much.  With its highly-specialized, technical and abstract content, math is usually not seen as a playground for non-professionals. Indeed many who try (and there ARE many) end up classified as "crackpots," their ideas or approaches so off-base and unworthy of attention.
In recent times there is the famous case of "homemaker" Marjorie Rice who, playing around with tessellations, made important contributions to the geometry of tilings. And there are a few others... but the number is small. In the last couple decades, how many "proofs" have come along for the Riemann Hypothesis or other "Millennium" problems from people dabbling way outside their competency, largely wasting time and energy.

I mention all this because of my ongoing fascination with the recent findings regarding the "pattern" of consecutive prime last-digits. Robert Lemke Oliver and Kannan Soundararajan who discovered it are professional mathematicians, but it is the sort of thing that could have been discovered by amateurs just 'playing around' with prime numbers (as people often do).  Indeed, a common response to their finding has been, 'HOW did this go UNnoticed so long!?" In this day of Mathematica and similar programs it seems the door is wide open to all manner of analyses of prime number digits/succession/position that amateurs could imagine doing -- it might well take a trained number theorist or other specialist to explain a given outcome, but just generating that outcome might be do-able by brute-force amateurs.

I've been following Mike Lawler, inspired by Oliver/Soundararajan's work, play with prime triplets for the last week with results that, while beyond my comprehension, could hold significance for others (Mike is not an "amateur," as he has a math PhD., but he is not a number theorist or prime specialist, and what he is doing could be done by a non-math PhD.). Current posts for his work are here:

https://mikesmathpage.wordpress.com/2016/03/27/my-fun-interaction-with-prime-numbers-this-week/

https://mikesmathpage.wordpress.com/2016/03/27/prime-triples-and-the-sieve-of-eratosthenes/

And he has been recording his results (looking at prime-last-digit-triples in billion increments) on an ongoing Google spreadsheet here:

https://docs.google.com/spreadsheets/d/1fZ0wkYrei3CR1XtUuqWKVubAbuM0EzFZ-SGOkKnapd0/edit#gid=0

This isn't for everyone, but for those mesmerized by the mysterious way prime numbers weave their way through our integer system, tantalizing us with their secrets and their non-random randomness(!), it can almost be addictive (beware). So here's to amateurs and professionals alike playing in that heady world of pure math, never quite knowing what they might find, where it might lead, or what it might mean.


...ADDENDUM:  after posting this, Mike put up another entry summarizing somewhat his experience thus far:
https://mikesmathpage.wordpress.com/2016/03/28/what-ive-learned-playing-around-with-primes/

Calling All Number Theorists....

Those persnickety primes... Earlier today Mike Lawler posted some results from Mathematica for the patterns in last digits of prime-triples, essentially in intervals of a billion primes (not sure if I'm stating that very clearly, but read his post):

https://mikesmathpage.wordpress.com/2016/03/24/weird-clustering-with-last-digits-of-3-consecutive-primes/

His columns are a bit hard to read, but you should be able to spot the "clustering" (or I would call it "coupling") he refers to which seems to largely hold for all 3 columns of the post. He notes in a comment that the "average" expected value for each entry would be about 15.6 million, so you can see how widely the values diverge from that, as well as see how they tend to pair up.
He is in the process of adding more columns at the below easier-to-read spreadsheet -- I assume he'll be going out to 10 billion primes, to restore data he originally had, but lost -- as I write this, columns for 4 billion primes are listed, and it appears to me (merely eyeballing it), that the numbers for the paired triplets are getting even closer(???): 

https://docs.google.com/spreadsheets/d/1fZ0wkYrei3CR1XtUuqWKVubAbuM0EzFZ-SGOkKnapd0/edit?usp=sharing

 As each set of a billion primes is a somewhat independent and random-like group of integers, this pattern of the same ordered-triplet of last digits re-occurring in associated pairs seems, on the surface at least, rather odd and striking!?  What (if anything) is it about those pairs? Perhaps a number theorist can see through to a simple explanation for it (if so, I'm sure Mike would love to hear it). Or does this finding piggy-back in any way on to the peculiar result from a week prior of prime number last digits tending to avoid repetition in consecutive primes?
WHAT is going on here....?

ADDENDUM:  I should have included in this post that Mike has already recognized that the paired triplets involved are consistently of the form (a, b, c) and (-c, -b, -a) in mod 10. Now THAT surely must mean something! (Again, perhaps something obvious to a number theorist, but WHAT?)