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!!!

Meditating... on Chaos

Food for thought… an interesting passage from chapter 85 (on chaos theory, also known as "bifurcation theory") of Richard Elwes' "Math In 100 Key Breakthroughs" (pgs. 345-7):

"How can one produce a random number? In the late 1940's, John von Neumann proposed a very strange answer to that question.  He suggested that applying a simple algebraic rule a few times should do the job. The rule is to begin with some number, call it x, and then multiply x by (1 - x), and multiply the result by 4. That is to say: x --> 4 X x X (1 - x).
"There does not seem to be anything especially 'random' about this bit of algebra. Once the initial number is chosen, say x = 0.1, the result of applying the rule is then completely predetermined. But a little experimentation reveals von Neumann's insight. The sequence produced by this rule runs: 0.1, 0.36, 0.9216, 0.2890, 0.8219, 0.5854, 0.9708, and so on (each number given to 4 decimal places). There does not seem to be much of a pattern here, and in fact that is no illusion. You can extend the sequence for as long as you like and in fact no pattern will emerge. Someone who did not know the rule being used would find it virtually impossible to distinguish between this sequence and one produced by a genuinely random physical process such as radioactive decay."…

"Today, von Neumann's rule is known as the logistic map, and it is one of the simplest examples of mathematical chaos, a phenomenon which has been recognized in many different situations…."

"In von Neumann's pseudorandom number generator, everything rests on the number 4, known as the parameter. Changing that value completely alters the behavior of the system. If one replaces 4 with a new parameter of 2, the logistic map ceases to be chaotic. Instead, for any starting value, the sequence will quickly home in on a fixed value of 0.5. This is known as an attracting point of the system.
"Increase the parameter from 2 to 3.4, and something new occurs. After a while, the sequence will endlessly flicker back and forth between two values around 0.84 and 0.45. This is known as an attracting 2-cycle. Raise the parameter a little higher to 3.5, and this is replaced with an attracting 4-cycle, and then at 3.55, an attracting 8-cycle, and so on. As the parameter increases, the length of the attracting cycle keeps doubling 15, 32, 64, and so on. This behavior is what chaos theorists call a sequence of bifurcations."

He goes on to explain that the bifurcations end once the parameter hits a certain threshold value known as the Feigenbaum point (named after chaos theorist Mitchell Feigenbaum). Beyond that point (like "4" in the example) the produced sequence will act chaotically forever, producing the famous "butterfly effect" whereby two sequences beginning at only slightly different starting values "end up entirely unrecognizable from each other."

A Big Serving of Monday Potpourri

Catching up on a few things:

1) First, Keith Devlin continues his fascinating series on his MOOC experience (now 6 recent posts to catch up on in his latest series, if you haven't been following along):

http://mooctalk.org/

2) And if you missed Keith delightfully talking on NPR this weekend about the excitement generated by Yitang Zhang's attack on the Twin-prime conjecture earlier this year, give that a listen here:

http://www.npr.org/2013/12/21/256003488/and-the-number-of-the-year-is-the-lowly-2?ft=1&f=1007

3) Meanwhile, Peter Woit writes about the incredible difficulty involved in verifying last year's proof from Shinichi Mochizuki of the ABC conjecture (perhaps unparalleled in the history of math):

http://www.math.columbia.edu/~woit/wordpress/?p=6514

4) And if anyone has missed it, my own latest interview with popular mathematician James Tanton is currently up at MathTango:

http://mathtango.blogspot.com/2013/12/james-tanton-making-math-accessible.html

5) Perhaps mathematicians are too shy or uncomfortable talking about their (or their colleagues') politics, but I'm still interested to hear from you, if you aren't:

http://math-frolic.blogspot.com/2013/12/are-mathematicians-liberals.html

6) And three for your mathematical entertainment:

a) Hat tip to The Aperiodical for pointing out this fun 30-min. BBC podcast on math and magic:

http://www.bbc.co.uk/programmes/b03ls7y2

b) Just this morning, Presh Talwalkar posted a nice triangle geometry problem with both a traditional algebraic solution and a 'quickie' simpler solution available:

http://tinyurl.com/mtovhpx

c) And finally, a link that got some play on Twitter last week is this old "urban legends" of math discussion from mathoverflow.com:

http://tinyurl.com/l6gapx6
(some interesting, fun, and quite technical 'urban legends' included...)

Are Mathematicians Liberals?

A speculative posting today… just something I'm wondering about:

I follow a LOT of professional (PhD.) mathematicians around the Web, and from their occasional dabbling in political/cultural matters, my strong impression is that the vast majority could be categorized as political "liberals." One of our icons, Martin Gardner, in his recent autobiography, unabashedly labeled himself a "democratic socialist" and cites Norman Thomas as one of his "heroes."
At first I thought this made simple sense (I mean after all, aren't all astute, thinking individuals, liberals ;-))) but then I began to wonder… most (though not all) of the mathematicians I follow are 'academic' mathematicians -- they have a working association with some academic institution. Perhaps it is the academic milieu that makes one liberal, moreso than the field of mathematics???
There certainly are 'professional' PhD. mathematicians who work for private industry; so I'm curious what their political leanings are, and if they differ much from the academic crowd. Any thoughts…?

If anyone cares to respond to any of the following questions, I'd be interested just out of curiosity (and of course you can be 'anonymous'):

1) First, does anyone disagree that the majority of academic mathematicians could be characterized as political liberals?

2) If you agree with that characterization, is there anything inherent to the advanced study of math that encourages 'liberalism' (or is it the result of completely separate factors)? i.e., does 'mathematical thinking' tend somehow to promote liberal thinking?

3) If YOU are a PhD. mathematician working in private industry (or know of some) do you find any significant differences in your political views and those of your academic colleagues? (and if so, any speculation on why that is?)

These are pretty wide-open questions and generalizations, so I don't expect precise, rigorous answers.
Also, I know there have been studies or surveys done of political attitudes broken down by professions; just don't recall if any have ever specifically included "mathematicians" as a category -- if anyone is aware of such a survey, available on the Web, let me know.

Hmmm… Probability Puzzles

 I just recently discovered this blog which appears completely devoted to probability puzzles... making me feel just a tad like Homer Simpson… IF you substitute the phrase "probability puzzles" for "beer" ;-) :

Yin and Yang… No, Zipf and Zhang

A couple of pieces to catch up on if you've missed them:

Yitang Zhang, the largely unknown, humble New Hampshire mathematician who sort of broke open the twin prime conjecture earlier this year, is now deservedly being awarded the Frank Nelson Cole Prize in Number Theory come January (he has already received the Ostrowski Prize). Read another lovely news story about this unforeseen math champion here:

http://www.unionleader.com/article/20131215/NEWS04/131219488

And the below article notes that Zipf's Law is finding more applications to human society than its original linguistic roots with which I was familiar (however, note that several commenters take issue with the article's author):

http://tinyurl.com/lakytzh

The Value of Failure

In his latest MOOC blog posts Keith Devlin insightfully discusses expectations, real math, mathematical thinking, AND, the 'power of failure.'
According to Keith high school math and college math are "in many ways completely distinct subjects," and the "mathematical thinking" needed at the college level cannot be "taught" but must be "learned," which includes "learning by failing":

"...it is only when we fail that we actually learn something. The more we fail, the better we learn; the more often we fail, the faster we learn. A person who tries to avoid failure will neither learn nor succeed."

Read his post (part 2) if you need clarification on all that -- heck read both posts (much food for thought) even if you need no clarification! -- at the risk of sounding like a broken record, these ought not be missed if you're an educator:

http://mooctalk.org/2013/12/14/maththink-mooc-v4-part-1/

http://mooctalk.org/2013/12/15/maththink-mooc-v4-part-2/

ADDENDUM (12/17): now part 3 of this series from Keith is up: 
http://mooctalk.org/2013/12/17/maththink-mooc-v4-part-3/

By the way, elsewhere, Dr. Devlin cites J.K. Rowling's 2008 commencement speech at Harvard (having to do with "failure") as possibly the best commencement address ever. If you've never seen it, enjoy:

Move Over, Clifford Pickover

Richard Elwes has a new volume out, "Math In 100 Key Breakthroughs," that I'd add to the Holiday math book shopping list I've already posted. It's a bit reminiscent of Cliff Pickover's "The Math Book" -- I like a lot of Pickover's stuff, and he was kind enough to do an interview for me here, but I was never greatly enamored of that particular volume from Cliff, despite its wide success and popularity -- I do however like Elwes' effort to combine math text and gorgeous graphics in a delicious way, that flows along nicely.

Elwes' book runs essentially in chronological order and while the first third didn't grab my interest that much, covering earlier math history, it gets more interesting with coverage of more modern mathematics (say starting with Newton onward). The text is again (like Cliff's book) on the pithy side, but a bit more substantive than the latter; and I always find Elwes to be one of the very best, clearest, current explicators of mathematical ideas for a lay audience… all the more reason I wish he had gone just a tad more deeply into some of the subjects addressed here.
Still, the volume represents, I think, a splendid introduction to the variety and range of mathematics, especially for a young person with such inclinations. It is already 400 pages long (perhaps at least 1/3rd of that from graphics/pictures), so maybe further, pedagogic text would've added too much. While organized into 100 chapters or "breakthroughs," each chapter covers multiple specific topics, so there's a lot more than 100 topics touched upon here, and more, I think, than is covered in Pickover's choppy volume of 250 "milestones."

My one beef with the book is that there is no bibliography included (Cliff's book has one at the end)… or even better yet, would have been a "for further study" listing following each chapter, referencing sources to further the reader's interest/knowledge if so inclined. One might argue that because anyone can Google any subject these days and find copious additional material, such bibliographic references are no longer needed… but it is exactly because Google returns such copious, ill-prioritized suggestions, that a honed list of excellent selections from the author would be valuable.

Anyway, I highly recommend this beautiful book, especially if you liked Pickover's more coffee-table-like version… OR, even moreso if you didn't find Pickover's volume satisfying, but still fancy the concept of combining wide-ranging, informative mathematical text with beautiful illustrations.

A > B > C > A

In time for the new year perhaps.…

You may be able to order a set of James Grime "nontransitive dice" (earlier sold out, but being re-supplied):

http://mathsgear.co.uk/collections/all-products/products/non-transitive-grime-dice

Nontransitive dice (also known as "Efron Dice" after one of the inventors) have probably regained some attention since being mentioned in Simon Singh's recent book, "The Simpsons and Their Mathematical Secrets." Several different nontransitive combinations are actually possible, but the set mentioned in Singh's book, include "Die A" with sides, 3,3,5,5,7,7, "Die B" with sides 2,2,4,4,9,9, and "Die C" composed of sides 1,1,6,6,8,8.  On average, a throw of Die A will beat (56% of the time) a throw of Die B, and a throw of Die B will beat (56% of the time) Die C… YET, Die C, on average, will beat out Die A (56% of the time)… How cool is THAT! or as, Singh writes, "Nontransitive relationships are absurd and defy common sense, which is probably why they fascinate mathematicians."

Wikipedia covers the subject here:

http://en.wikipedia.org/wiki/Nontransitive_dice

But of course James Grime is far more entertaining here:

http://singingbanana.com/dice/article.htm

Fans of the simple competitive game "Rock, paper, scissors," may recognize that it too operates on the basis of nontransitivity.


On a sidenote, the current post over at MathTango contains some puzzle fun, including my re-tell of an old favorite from Raymond Smullyan:

http://mathtango.blogspot.com/2013/12/puzzles-puzzles.html

AND finally, a reminder that Sunday will be the last day of the sparsely-entered :-( caption contest over at MathTango:

http://mathtango.blogspot.com/2013/11/caption-contest.html

Why oh Why?

"Why Write a Math Blog" -- that was the title of a recent short post at Ken Abbott's blog, where he offered 3 reasons for his blogging:

http://www.math-math.com/2013/11/why-write-math-blog.html

His thoughts inspired me to think about my own reasons for math-blogging, since my sparse (academic) math background makes it an even more interesting question for me… and my reasons overlap, but differ a bit, from Ken's.
Indeed, when friends have asked, in a surprised tone, "why" do I math blog my first reaction is to explain that while I write ABOUT math, I don't actually DO much mathematics on the blog. I'm more interested in the topic of math and mathematicians than in the working out of math, for which I have limited competency.

So reasons for math-blogging here at Math-Frolic are:

1)  This blog was born with the demise of Martin Gardner back in June 2010 -- I was doing a small science blog at the time, but Gardner's death brought back memories of the hours of enjoyment I got from his work, and I decided to begin anew with a blog focusing on math in his honor. (expecting it to be short-lived, but over time it grew, as to my naive amazement there seemed to be never-ending material to draw from.)

2)  I like trying to make math interesting to others.

3)  I enjoy curating information, and writing.

4)  In the process of doing the blog I get to learn a great deal myself.

5) And best of all, mathematicians are among the most interesting people in the world to me, and the blog puts me in contact with people I would otherwise never have had the pleasure of crossing paths with!

So HOORAY to the world of math-blogging! (I'm sure other math-bloggers have their own reasons for blogging, sometimes different from the above -- feel free to chime in with your own motivations/rewards in the comments).

Excitement... from 1984

The math bibliophile in me is always excited when our local public library holds a used book sale... and today my lucky find was "Mathematics: People Problems Results," a 3-volume anthology set from 1984, edited by Douglas Campbell and John Higgins. It looks to be a scrumptious set of ~90 rich essays (including some classics) from a great panoply of superb (and famous) writers/mathematicians. As I've said before, math is so timeless that even a 30-year-old book-set like this can contain fabulous stuff.

This all reminds me that the current caption contest over at MathTango only has two entrants thus far and I'll probably end it next Sunday (the 15th), so give those folks some competition and get your entries in.

Singh, Simpsons, Science Friday

If you missed Simon Singh (author of "The Simpsons and Their Mathematical Secrets") and David Cohen on NPR's "Science Friday" today with Ira Flatow, well, you're in for a treat:

http://tinyurl.com/kdys4wz

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.]

But Tell Us How You Really Feel, Dr. Z

Communicating math… aye, there's the rub:

Rutgers professor Doron Zeilberger has a bit of a gadfly/curmudgeon reputation in the mathematical community… which is what (in part) makes him so interesting to hear out! He has an opinion piece about math communication in the latest "Notices of the AMS" which is getting some buzz, including inspiring a blog post from Jason Rosenhouse. The Zeilberger letter (pdf) is here:

http://www.ams.org/notices/201311/rnoti-p1431.pdf

In it, he criticizes "pure mathematics" for its 'fanatical' focus on "rigorous proofs," and urges greater emphasis on "experimental mathematics" noting:

"Mathematics is so useful because physical scientists and engineers have the good sense to largely ignore the 'religious' fanaticism of professional mathematicians and
their insistence on so-called rigor, which in many cases is misplaced and hypocritical, since it is based on 'axioms' that are completely fictional, i.e., those that involve the
so-called infinity.
"The purpose of mathematical research should be the increase of mathematical knowledge, broadly defined. We should not be tied up with the antiquated notions of
alleged 'rigor'."

It is an interesting (recommended) read and not very long, and his criticism of higher-level math communication as "highly dysfunctional" (only comprehensible to the few specialists who share a given area of work), spurred Jason Rosenhouse to write his own interesting blog post, largely in agreement:

http://scienceblogs.com/evolutionblog/2013/12/03/communicating-mathematics/

Rosenhouse's piece would be worthwhile alone for his own illustrative account of a math conference presentation as a nervous grad student; but there are many other good points in the post as well.

I recommend you read both these pieces, and you may want to also check out some of Dr. Zeilberger's other (wishy-washy, NOT!)  opinions here:

http://www.math.rutgers.edu/~zeilberg/OPINIONS.html

The Oracle of Devlin ;-)

I wish that just once Keith Devlin would write a blog post that I could yawn at and didn't feel obligated to refer my readers to. But the man just seems incapable of writing anything mundane or trite or ordinary. His latest thoughtful offering, on MOOCs and "quantitative reasoning," here:

http://devlinsangle.blogspot.com/2013/12/moqr-anyone-learning-by-evaluating.html

I love watching Dr. Devlin's experience with MOOC-building evolve over time, and his openness/honesty in letting us observe as he rides the roller-coaster of hope/doubt/optimism/pessimism/confidence/uncertainty that seem to coincide with the development of MOOCs (if not education change/reform in general!!)
He will be substituting something he calls "Test Flight" in place of a final exam in the next iteration (beginning Feb. 3) of his own mathematical-thinking MOOC, and watching to see if it succeeds or 'crashes and burns.'

He winds down this particular piece with these contemplative words:

"The more people learn to view failure as an essential constituent of good learning, the better life will become for all. As a world society, we need to relearn that innate childhood willingness to try and to fail. A society that does not celebrate the many individual and local failures that are an inevitable consequence of trying something new, is one destined to fail globally in the long term."

A Stocking Stuffer, Perhaps

On impulse about a month ago (and trying to use up a discount-coupon ;-), I purchased a little math reference volume at my local Barnes and Noble, entitled "Math In Minutes: 200 Key Concepts Explained in an Instant" by Paul Glendinning (or "Maths In Minutes" for the British version). Turns out it was published in 2012, even though I didn't see it 'til a few weeks back (as a British offering it may have taken awhile to reach the States).
Anyway, wasn't planning to mention it here, but occurs to me it might make an okay stocking stuffer for some budding math person on your holiday shopping list so I'll give it a plug. At about 5"x 5"x 1" it will literally fit in some oversized fireplace stockings! Amazon describes it, in part, thusly:

"...simple and accessible... introduction to 200 key mathematical ideas... described by means of an easy-to-understand picture and a maximum 200-word explanation… Compact and portable format -- the ideal, handy reference."

"Ideal" is probably too strong a word, but definitely "handy." The format actually is reminiscent (in miniature) of Clifford Pickover's wildly-popular "The Math Book," in so much as there is generally a brief text on the left-hand page followed by a pertinent (black-and-white) picture on the right-hand page. Not as glossy or beautiful as Cliff's work, and the text is even thinner (indeed, rather superficial) than Pickover's, but the trade-off is a very portable, bite-size volume, that still touches a lot of ground. Also, unlike Pickover's strictly chronological format, the Glendinning offering categorizes its 200 ideas into broader topic areas:

- Numbers
- Sets
- Sequences and series
- Geometry
- Algebra
- Functions and calculus
- Vectors and matrices
- Abstract algebra
- Complex numbers
- Combinatorics
- Spaces and topology
- Logic and proof
- Number theory

Again, it's not a great book you need to rush out and get for your own shelf, but perhaps a fun little reference/gift for some young mathematician you want to surprise.

A Few Quotes

Just a few quotations for your reflection today, taken from Simon Singh's "The Simpsons and Their Mathematical Secrets":

From famous British mathematician G.H. Hardy: "Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. 'Immortality' may be a silly  word, but probably a mathematician has the best chance of whatever it may mean."

From Simpsons writer (and Harvard physics grad) David Cohen, on the satisfaction derived from slipping mathematics into Simpsons' episodes: "I feel great about it. It's very easy working in television to not  feel good about what you do on the grounds that you're causing the collapse of society. So, when we get the opportunity to raise the level of discussion -- particularly to glorify mathematics -- it cancels out those days when I've been writing those bodily function jokes."

And from author Singh: "It would be easy for non-nerds to dismiss the mathematical shenanigans that appear on The Simpsons and Futurama as superficial and frivolous, but that would be an insult to the wit and dedication of the two most mathematically gifted writing teams in the history of television. They have never shied away from championing everything from Fermat's last theorem to their very own Futurama theorem.
"As a society we rightly adore our great musicians and novelists, yet we seldom hear any mention of the humble mathematician. It is clear that mathematics is not considered part of our culture. Instead, mathematics is generally feared and mathematicians are often mocked. Despite this, the writers of The Simpsons and Futurama have been smuggling complex mathematical ideas onto prime-time television for almost a quarter of a century."