John Baez Recounting Ramanujan

 Anecdotes and speculation about anecdotes (the ever-interesting life of Ramanujan):

https://johncarlosbaez.wordpress.com/2022/01/30/hardy-ramanujan-and-taxi-no-1729/

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.

RFI… Can Any Number Theorist (or others) Reply to This?

I can’t answer the following for one of my readers, but maybe someone else can (about a Fermat factoring variant)???

Well over 3 years ago I posted this little gem from Futility Closet:

"In 1643, Marin Mersenne wrote to Pierre de Fermat asking whether 100895598169 were a prime number.

Fermat replied immediately that it's the product of 898423 and 112303, both of which are prime.

To this day, no one knows how he knew this. Has a powerful factoring technique been lost?"

At the time, one comment came in (from someone named Walt), as follows:

“It is not that surprising if we assume that Fermat used what we call Fermat's factoring method. Upon multiplying by the cofactor of 8 (and using difference of pronic numbers instead of squares since the number is now even) he would find this pair of factors on the first try. (I am assuming that taking the square root of a 12-digit number was feasible. Also, I have no idea how difficult it was at that time for him to prove the primality of the resulting 6-digit factors.)
In fact, 8 * 112303 = 898424 = 1 + 898423. This is a remarkable coincidence and makes me wonder if Mersenne used this relation to construct the problem in the first place.”

Now (3+ years later) I’ve received an inquiry from a retired German mathematician, who recently ran across the post & comment, and asking in part about the:

“…. variant of ‘what we call Fermat's factoring method. Upon multiplying by the cofactor of 8 (and using difference of pronic numbers instead of squares since the number is now even) …’  I am wondering about (t)his remark to use differences of pronic numbers: I have never heard of that variant or read about it in any book. He [Walt] does not quote any references, so he seems to consider this variant as being well known. In fact, it is not too difficult to figure out the formula a*(a+1) - b*(b+1) = (a+b+1)*(a-b) that turns such a difference into a product, and to check that one may thus devise a factorization method for even numbers. I am wondering if this has been published anywhere.”

Can anyone provide an answer???

"an extreme sensitivity to numbers"

Am largely taking vacation from blogging for a couple of weeks (’til political conventions are over), and will again skip Friday potpourri over at MathTango, but put up an occasional post here (and still be on Twitter) -- between the two blogs I've averaged over 5 posts per week for the last 6 years so won’t feel too guilty taking a vacation ;-)

Anyway, passing along this interesting recent piece from the Christian Science Monitor on a supposed real life “Good Will Hunting” Chinese migrant (Yu Jianchun) using a creative/imaginative approach to solve a long-standing problem involving “Carmichael numbers” :

http://www.csmonitor.com/Science/2016/0719/In-China-a-real-life-Good-Will-Hunting

…alternatively, this coverage from the Washington Post:

https://www.washingtonpost.com/news/morning-mix/wp/2016/07/19/parcel-delivery-man-says-hes-solved-tough-number-problem-heralded-as-chinas-good-will-hunting/

With all the reporting on Ramanujan in recent months (including in these articles), Yu's story sounds a bit familiar. According to one professor, “All he has is an instinct and an extreme sensitivity to numbers.” And Yu himself says, “I made my discoveries through intuition.”

I love this almost inexplicable notion of math prodigies and savants possessing a “sensitivity to numbers,” whatever that means, and connecting to mathematics more through intuition than pure deduction. In some way it harks back to the Platonist/non-Platonist divide in mathematics. Are such gifted individuals intuitively in touch with some Platonic realm of math that exists apart from humans, and that most of us lack direct access to, or are they merely in touch with some special corner of their own working brains? Are they discovering math or creating it? And what is it like to be “sensitive” to something as abstract and ethereal as numbers?

It all makes me think a bit of physicist Max Tegmark's controversial view that all there is in the Universe is mathematics (or mathematical structure), and ultimately nothing more. But then how would such mathematical structure evolve into human brains capable of looking back on itself with objective analysis? And are the philosophers and cognitive scientists who tackle such questions simply caught in some sort of infinite regress or word loop... explaining an explanation by an explanation of an explanation of... (that really explains nothing!).
Anyway, go read about the "package delivery worker" Jianchun who, after 8 years of emailing prominent mathematicians "to no avail," finally got someone to take note.

A Name to Know; Work to Be Aware Of

From Erica Klarreich at Quanta, a fascinating piece about a fascinating young mathematician, his fascinating work in number theory, making fascinating, groundbreaking connections between disparate areas of math:

https://www.quantamagazine.org/20160628-peter-scholze-arithmetic-geometry-profile/

Did I mention this is fascinating stuff....

Peter Scholze is a 28-year-old German wunderkind, probable Fields Medal candidate, and by several accounts, "one of the most influential mathematicians in the world," who works at the intersection of number theory and geometry. That might sound simple, but it is cutting edge, and for most, unexplored territory. Yet Peter seems to possess a strong intuitive sense for it (the article is aptly titled, "The Oracle of Arithmetic"). 

A couple of quick sentences from the piece:
"'I’m interested in arithmetic, in the end,' he [Scholze] said. He’s happiest, he said, when his abstract constructions lead him back around to small discoveries about ordinary whole numbers."

Almost makes it sound as if we rookies could understand what he does ;-); but that'll be the day. He's delving into deep, rich, abstract areas of mathematics, that most of us will never encounter, but the article makes clear he is also open, generous, and patient in his willingness to explain it to those who are able to take the leap.

Klarreich writes that Scholze "avoids getting tangled in the jungle vines by forcing himself to fly above them," which reminded me so much of Keith Devlin's early metaphor of reaching the top of a mathematical woodland canopy where he could look down and suddenly see that the whole forest was inter-connected.
Part of Scholze' work deals with what is called "reciprocity" and its linkage to hyperbolic geometry, including "perfectoid spaces," all of which leads to the Langlands Program and "frontiers of knowledge" which may eventually unify the field of mathematics (slightly akin to the so-called "Theory of Everything" searched for in physics).

But I can't do Scholze or Klarreich's writing justice here, so go read her article NOW!

Don Knuth Explains the Surreals

Wonderful new video from Numberphile, of Donald Knuth describing where "surreal numbers" came from:



If you missed it, less than a year ago Jim Propp ran this great post on the surreals at his MathEnchantments blog:

https://mathenchant.wordpress.com/2015/08/12/the-life-of-games/

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

Prooooooooooof

A great followup today from John Baez to the 200-terabyte "very long proof" story Evelyn Lamb reported on last week:

https://johncarlosbaez.wordpress.com/2016/05/28/very-long-proofs/

At one point John writes, "It’s interesting that these 200 terabytes were used to solve a yes-or-no question, whose answer takes a single bit to state: no."  ;-)

I don't know what application this may all have (???), but fascinating stuff nonetheless!

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?)

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)

Laysplaining, Mathsplaining, and Weeds...

When I wrote my Master's thesis a few eons ago, for fun I slipped in a few casual, informal bits... which my adviser saw and asked, "You weren't planning to leave that in the final draft were you?" To which I responded, "Well, actually, yes; you know, just trying for a little levity and less stodginess." And he said, "You can't do that." Needless to say, the final version reverted to academese.

I was reminded of that long-ago episode after Jordan Ellenberg tweeted out a link this week to the below math thesis which describes itself as "a fascinating tale of mayhem, mystery, and mathematics." It's been buzzing around the intertubes ever since, and may just become THE most viewed math dissertation in history!:
 https://twitter.com/JSEllenberg/status/674245895580426241

It hails from Princeton graduate Piper Harron, and the original (more academic) version of the material was posted on arXiv a couple years back:
http://arxiv.org/abs/1309.2025

There's already been a lot of commentary about the dissertation on the Web. Among my favorite remarks was this:

"I don't know enough about higher math to evaluate her work, but I can tell she's absolutely brilliant. Because you have to be brilliant to get away with that amount of sheer attitude."

Indeed, I've also seen some quite negative commentary... emanating from folks I suspect are lacking in appreciation for humor, creativity, and certain attitude! (there's no real reason that math, even pure math, can't include those).

The actual mathematics involved may weight you down, so try to stay focused on the larger storyline/ideas Piper is conveying. A few lines from the "Prologue" to get you started:

"Respected research math is dominated by men of a certain attitude. Even allowing for individual variation, there is still a tendency towards an oppressive atmosphere, which is carefully maintained and even championed by those who find it conducive to success... My thesis is, in many ways, not very serious, sometimes sarcastic, brutally honest, and very me. It is my art. It is myself. It is also as mathematically complete as I could honestly make it...
"It is not my place to make the system comfortable with itself. This may be challenging for happy mathematicians to read through; my only hope is that the challenge is accepted."

...and perhaps then too, keep in mind the old saying, "Attitude is everything!" ;-)


ADDENDUM:  the inimitable Mathbabe (Cathy O'Neil) now has a guest post up from Piper herself further explaining her "thesis grenade":
http://mathbabe.org/2015/12/11/piper-harron-discusses-her-artistic-and-wonderful-math-ph-d-thesis/

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."

Lychrel Numbers

A Lychrel number is an integer (any length) that does NOT eventually form a palindromic number (reading the same backwards & forwards) after following a process of reversing its digits and adding, reversing and adding, etc....
for example, starting with 837:

837          1575        7326        13563         50094
738          5751        6237        36531         49005
1575        7326      13563        50094         99099      <== palindromic

Simple enough, and in most cases a palindrome is arrived at within fairly short order.
BUT in the case of 196 no such palindrome has ever been reached. There is no proof that one does not exist somewhere waaaaay out there, though it seems unlikely; but why? why 196? There are actually several additional numbers thus far not found to produce palindromes, 196 is just the smallest of them. In fact, NO Lychrel numbers have ever been proven to exist in base 10, just plenty of candidates.

More from MathWorld: 
http://mathworld.wolfram.com/196-Algorithm.html

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!

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

Weird Math...

What do the two numbers, 70 and

26,963,672,211,957,831,828,322,834,071,143,299,817,754, 720,290,127,404,079,937,026,385,368,922,075,196,690,720,690,562,498,337,038,657,263, 353,255,952,256,005,850,803,053,091,152,216,128,172,198,270,512,414,580,092,743,322, 379,544,478,286,025,897,899,890,351,444,085,611,625,835,160,270,418,964,124,507,243, 890,975,821,522,176,465,361,680,177,670,297,930,314,037,850,339,675,559,057,554,452, 347,547,946,165,134,639,879,111,112,583,151,946,671,967,876,920,506,598,818,088,728, 910,330,021,016,856,674,391,763,268,224,262,067,132,913,691,721,407,174,127,885,521, 288,146,239,271,038,154,486,086,650,600,357,888  ...have in common?

They are both "weird" numbers… and I mean that, in a technical way! 70 is the smallest "weird" number and that second monstrosity is, to date, the largest known (of an infinite number) of weird numbers, at 226 digits. It was found by these Central Washington University folks:

http://www.kimatv.com/news/local/CWU-math-students--234496131.html
also see here: http://tinyurl.com/knzfswc

"Weird" numbers are those natural numbers whose divisors add up to more than the number itself, and for which NO selection of divisors sum exactly to the original number [for example, for 70, the divisors are 1, 2, 5, 7, 10, 14, and 35, which sum to 74, and no possible combination adds exactly to 70]. The student group originally discovered the first new weird number in over three decades, with a 72-digit find, before eventually reaching the above record.  Per the article, "a better understanding of weird numbers leads to a better understanding of factorization, which is the basis of all modern cryptography." [in case you were wondering of what possible use this could be!]

Here is a sequence of weird numbers from the OEIS directory:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670

Interesting that all of these, with the single exception of 836, end with a "2" or a zero, yet the new record find ends with an 8. -- I have no idea what the distribution of end-digits is for the full panoply of currently-known weird numbers??? (It is also not known with certainty if ANY odd weird numbers exist... but if they do, they must be very, VERY large!)

[I don't know if it's even possible to explain at a layperson level, but if someone in-the-know wants to try and explain in the comments what sort of method/algorithm one employs to discover weird numbers of such length (or alternatively how one verifies such a number) I'd be curious to hear it.]

Saturday Potpourri...

A few sundry items for your artful attention and possible perusal ;-):

1) Lance Fortnow, computer scientist and author of "The Golden Ticket," (which I reviewed a bit ago), all about P vs. NP, is Sol Lederman's latest podcast guest at Wild About Math:

http://wildaboutmath.com/2013/04/12/lance-fortnow-inspired-by-math-28/

2) A nice little primer on the nature of real numbers and pi from physicist Matt Springer here:

http://scienceblogs.com/builtonfacts/2013/04/12/everything-in-pi-maybe/

3) An interesting-looking list here of 24 video lectures in number theory:

http://www.infocobuild.com/education/learn-through-videos/mathematics/introduction-to-number-theory.html

4) Just a heads-up that E.O. Wilson is scheduled to be on NPR's Sunday "Weekend Edition" (tomorrow). I assume there will be some discussion of his recent much-debated commentary asserting that scientists need not know advanced mathematics to be successful.

5) Finally, a site I only recently learned of called "Ideas Roadshow" which looks interesting and includes this recent 5-minute clip by philosopher James R. Brown on Platonism in mathematics:

http://www.ideasroadshow.com/issues/james-robert-brown-2013-04-12

33,550,336 is a Perfect Number

...It is one of just 48 that are known (the last one just discovered a couple months ago). Are there any odd ones? Are there an infinite number of them? No one knows for sure.

Read about 'perfect numbers' from this Mario Livio post for Huffington Post Science:

http://www.huffingtonpost.com/mario-livio/perfect-numbers_b_2998917.html

Will Another Conjecture Bite-the-dust?

Peter Woit reports the claim that the "abc conjecture" has been proven:

http://www.math.columbia.edu/~woit/wordpress/?p=5104

The abc conjecture is a number theory conjecture from 1985 which has been called "the most important unsolved problem in Diophantine analysis."

Peter links to this more detailed report and discussion of the claim:

http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/

and here's a more layman-friendly interpretation of it from a few years back:

http://bit-player.org/2007/easy-as-abc