Those Befuddling Primes...

 Jim Propp muses about the nature of the primes:

https://mathenchant.wordpress.com/2022/02/16/the-clatter-of-the-primes/

Twin Primes and Their Shared Neighbors

 “It’s as if the primes were heaving all their divisors over the fence into the neighbor’s yard”….

Another splendid post from Brian Hayes (this time on prime numbers & “tweens”):

http://bit-player.org/2021/does-having-prime-neighbors-make-you-more-composite

Probably room for a lot more work on this... primes are endlessly fascinating.

There Are Proofs, and, Then There Are Proofs...

 This is above my pay grade ;) but some of you may likely find this Gil Kalai piece stimulating:

https://gilkalai.wordpress.com/2021/04/22/the-probabilistic-proof-that-2400-593-is-a-prime-a-revolutionary-new-type-of-mathematical-proof-or-not-a-proof-at-all/

Prime Stuff...

Always elusive prime numbers...

1) Super piece from Kevin Hartnett in Quanta today about the work of Kaisa Matomäki on prime factors and related ideas:

https://www.quantamagazine.org/kaisa-matomaki-dreams-of-primes-20170720/

2) ...and timely, as it follows up on a new Numberphile video yesterday with James Maynard on prime gaps:

3)  And earlier in week Evelyn Lamb pointed out this fun li'l excursion into prime numbers I'd missed from a few weeks back:
http://mathmisery.com/wp/2017/06/25/summer-excursion-1-prime-numbers/

No Largest Prime Gap

I've reported on this in the distant past, but since Mike Lawler recently asked bloggers to post some entries that might be of interest to both mathematicians and students, I’ll re-run this simple, old demonstration that you can have ANY size gap between two prime numbers that you want. I’ve always liked it, for its simplicity, since first seeing it in a popular 1984 volume from Laurie Buxton called “Mathematics For Everyone.” It runs like this (using Buxton’s example):

Hopefully you know what 600! means, i.e. the product of 600 x 599 x 598 x ….. x 2 x 1.

A pretty large number, but we need not actually multiply it out. Now consider the following string of consecutive numbers:

600! + 2

600! + 3

600! + 4

600! + 5

.

.

.

600! + 600

The above produces a list of 599 consecutive integers, NONE of which can be prime. Every number here will be divisible by at least the number on the right (because 600! is divisible, without remainder, by every number UP TO 600, and adding anything between 1 and 600 simply includes one of those divisors). Thus, in this example we have a gap of at least 599 integers without a prime appearing. BUT clearly one need not start with 600. One can start with a number as large as one likes in order to generate a prime gap as large as one wants. There will never be a largest gap. Simple and convincing!

Zhang and the Twin Primes...

If you've never seen this 2015 documentary (uploaded to YouTube about a month ago) on Yitang Zhang and his Twin Prime Conjecture work it's worth a watch:



It's almost an hour long (starting off with fairly basic information) and very enjoyable, both for the remarkable storyline and for the appearances of lots of key people.

Playing With Primes...

James Maynard continues his fascinating work with prime numbers:

https://plus.maths.org/content/missing-7s

Maynard has shown that there are an infinite number of primes that include no number “7” digit. As the article states:

" ‘The vast majority of big numbers have lots and lots of 7s in them, so having no 7s is a rare property for any whole number to have.’ The fact that, despite this rarity, Maynard was able to prove that there are infinitely many such primes counts as an impressive feat. There's nothing special about the number 7, incidentally. Maynard's proof works equally well for any other number: so we now know that there are infinitely many primes without 1 as a digit, or 2 as a digit, or 3, or 4, or 5, and so on.”

The article proceeds with some interesting exposition about primes, music, Fourier analysis, “waves,” and cuboids.

But returning to the digit analysis I can’t help but wonder if there are also an infinite number of primes missing any two digits, say “7” and “3” for example, or “1” and “2”… or of course then three digits… or…. (what are the limiting cases here, and what, if anything, would they mean?). Or, starting at the other end, are there an infinite number of primes composed of say, nothing but 1s or 7s  (DOHH!!, that wouldn't work).

I don’t see any indication in the piece how much of this has already been looked at? HEY, Mike Lawler, a weekend project! ;-)

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:

http://bit-player.org/2016/prime-after-prime

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?)

The March of Primes

               ...1     ...3     ...7      ...9


Last week when an article reported looking at 'patterns' in the last digits of consecutive primes, I mentioned I thought it swung the door wide open to a plethora of prime digit data exploration one might do. Mike Lawler gets that ball rolling in this post (and I suspect others out there are looking at a variety of possibilities):

https://mikesmathpage.wordpress.com/2016/03/20/the-last-digits-of-triples-of-consecutive-primes/

(specifically, -- Mike looks at final digits of prime triples, but one can imagine plenty of other possibilities -- having said that, it's also possible that the sheer act of looking over loads of data, now so easy to generate, will result in occasional pattern-like findings emerging... that may lack any real meaning, beyond "chance").
Anyway, could all make for a very interesting year ahead in primes and number theory....

Story of the Week! (and it's only Tuesday)

Like a good magician, prime numbers never quite reveal everything held up their sleeves.
As most have likely heard already the momentous story of the week (perhaps the month) for many of us, doesn't include Donald Trump, but rather a new 'pattern' or bit of non-randomness noted for prime numbers.

Of course primes aren't truly random to begin with (in their distribution throughout the number system), but this finding indicates that even their final individual digits appear for some reason skewed away from randomness, with a given prime being followed by another prime who's final digit differs from the first with greater-than-expected probability. As articles have mentioned, it is remarkable it's taken this long for anyone to notice. Like the magician who's sleight-of-hand distracts us from seeing what's right in front of our eyes. Whether this finding will have any practical application is difficult to see; for now it simply sits baldly and boldly in the rarefied domain of number theory awaiting further explication.

Significantly (I would think) it may open up a Pandora's Box of other questions to be looked at regarding individual digits of primes and their relationships in placement, order, succession... and will any variances from 'randomness' discovered be mere mathy statistical glitches or 'accidents', or do they hold some rich, deeper meaning not yet understood? I suspect most believe the latter.

Anyway, the story started in the popular press (so far as I'm aware) with Quanta Magazine and this fantastic piece by Erica Klarreich:
https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/

Evelyn Lamb covered the subject for Nature:
http://tinyurl.com/zzj2yd6

ADDENDUM:  Dr. Lamb now also has this fascinating followup post at her "Roots of Unity" blog
http://blogs.scientificamerican.com/roots-of-unity/two-plausible-facts-that-cannot-both-be-true/

John Baez offers good coverage here:
https://golem.ph.utexas.edu/category/2016/03/unexpected_biases_in_the_distr.html#more

And more technically, Terry Tao here:
https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes/

(there are many other articles, but these give a great run-down)

Producing Those Pesky Persnickety Primes...

5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3 . . .  Rowland's sequence


The always-interesting Brian Hayes takes readers on a rollicking journey with formulas created to produce prime numbers:

http://bit-player.org/2015/pumping-the-primes

Both computer programmers and number theorists may find this interesting, even though in the end, Hayes admits that "It seems we are back where we began, and no closer to having a practical prime generator"...but, as he also concludes, "Along the way you may have seen something interesting, or even astonishing."

Primes, Zeta, Riemann, Oh My (from Marcus du Sautoy)

Love this podcast (~28 mins.) with Marcus du Sautoy on prime numbers, the zeta function, Riemann Hypothesis, etc.:

Quantum Physics and Pure Math Creeping Toward One Another

Seed Magazine has just re-run a fascinating 2006 piece from Marcus du Sautoy on a possible deep connection between the mystery of prime numbers and the quantum physics of atomic structure:

http://seedmagazine.com/content/article/prime_numbers_get_hitched/

It relates back to one of my favorite 'Sunday reflections' here involving a famous encounter between Freeman Dyson and Hugh Montgomery.

Here is a more recent longish piece (2013) on the same subject from Princeton's Institute for Advanced Study that brings quasi-crystals into the discussion:
https://www.ias.edu/about/publications/ias-letter/articles/2013-spring/primes-random-matrices

And a whole lot more from this 2013 John Baez posting over at n-Category Cafe blog:
https://golem.ph.utexas.edu/category/2013/06/quasicrystals_and_the_riemann.html
(again, technical stuff!)

Matthew Watkins' trilogy of books on prime numbers, for a more general audience, also relates to this discussion.

One gets the feeling there is something very fundamental going on here... verrrrry fundamental... but, can the human brain unravel it!?

NOT To Be Missed... on number theory/prime gaps

"After a while, these things taunt you".... (T. Tao)

FANTASTIC piece from Erica Klarreich and Quanta Magazine today on another obvious, but deep question from number theory (how LARGE can prime gaps be? ...sort of the reverse of the twin-prime question):

https://www.quantamagazine.org/20141210-prime-gap-grows-after-decades-long-lull/

Includes a "favorite joke" of number theorists that I'd not heard before :-) and also perhaps my favorite photo from all of mathematics: Paul Erdös and Terence Tao (as a child) together.
Seriously, with mentions of Yitang Zhang, Erdös, Tao, James Maynard, prime gaps, a crazy-ass log formula, and $10,000 prize, what is there not to love!

Short Film About Prime Numbers

 Kinda coool! (h/t to Gary Davis for pointing to this):



You can read more about it here (with several more links):

http://fadesingh.tumblr.com/post/91837545374/prime-numbers-magic-cicadas-and-mexican-bus-drivers

Stats and Primes...

Today, a couple of statistics-related readings from the latest John Brockman/Edge volume, "What Should We Be Worried About?"

1) Bart Kosko on what he deems five "lamplight probabilities" that help explain the world we observe, but also limit our analysis, and 'especially restrict modern Bayesian inference':

http://edge.org/response-detail/23856

2) And Nassim Taleb talking about one of his favorite subjects, "fat tails," and why "having skin in the game" is a necessary component for honest, accurate measurement of risk:

http://edge.org/response-detail/23839

...In other matters, an update on the twin-prime conjecture, now down to an upper bound of 252 (from the original approach based on Zhang's work):

http://tinyurl.com/m3pt98s

Prime… Yes or No

Returning from their excursion into the addition of infinite series, to just plain ol' normal levels of incredibleness ;-) Numberphile is back with the AKS primality test (initials from the names of its three originators), which amazingly, is an algorithm discovered only in 2002, for testing whether or not any integer is a prime. There are some other similar algorithms, but as Wikipedia states, "AKS is the first primality-proving algorithm to be simultaneously general, polynomial, deterministic, and unconditional." And from Wolfram MathWorld this Paul Leyland quotation regarding the finding: "One reason for the excitement within the mathematical community is not only does this algorithm settle a long-standing problem, it also does so in a brilliantly simple manner. Everyone is now wondering what else has been similarly overlooked."
Anyway, watch and enjoy as James Grime explains further:




ADDENDUM:  just discovered that Grey Matters blog has also done a nice explanatory post on the Numberphile video here:

http://headinside.blogspot.com/2014/02/aks-meets-pascal.html

Prime Progressions… (by 544,680,710)

Following up on yesterday's words from Richard Elwes I'll toss out another bit from his wonderful volume "Math In 100 Key Breakthroughs" (pg. 386). We're all familiar with "twin primes" like 11 & 13, and even triplet primes like 3, 5, 7. One can also signify longer sequences of primes that are separated by equal gaps: 11, 17, 23, 29, for example have spacing 6-apart (there are other primes, 13, 19, interspersed, but we're ignoring them).

Richard Elwes asks, "How long can sequences like this be?" and replies further, "The search quickly becomes hard, as the individual numbers involved become very large, too. The longest currently known arithmetic progression of primes consists of 26, beginning with 43,142,746,595,714,191 and then increasing in steps of 544,680,710. It has long been conjectured that there should be arithmetic progressions of primes of every possible length. This idea dates back at least to 1770, to the work of Edward Waring and Joseph Louis LaGrange. But the conjecture resisted all attempts at proof until 2004, when Ben Green and Terence Tao collaborated to prove their stunning theorem.
"If you want a list of 100 primes, each exactly the same distance from the last, the Green-Tao theorem guarantees there will be such a list somewhere. It does not, however, provide much useful information about where to start looking!"

...mind… blown… yet… again. . . .

…and, as long as we're speaking about primes, I hope most of you saw Web cartoonist xkcd's recent effort on the Goldbach conjecture(s):  http://xkcd.com/1310/

Finally, a safe, happy... and mathy NEW YEAR to one-and-all!!!