This month's topics:

The longest proof ever

Evelyn Lamb has a news report in Nature (May 26, 2016): "Two-hundred-terabyte maths proof is largest ever," with the subtitle: "A computer cracks the Boolean Pythagorean triples problem--but is it really maths?" As Lamb explains, "The problem asks whether it is possible to colour each positive integer either red or blue, so that no trio of integers $a, b$ and $c$ that satisfy Pythagoras' famous equation $a^2 + b^2 = c^2$ are all the same colour. For example, for the Pythagorean triple 3, 4 and 5, if 3 and 5 were coloured blue, 4 would have to be red." The answer is no, and the proof by Marijn Heule, Oliver Kullmann and Victor W. Marek, submitted to arXiv on May 3, 2106, shows that even though such a coloring is possible for all integers up to 7,824 it cannot be extended to 7,825.

corner of array

The lower left-hand corner of the $100\times 79$ array of squares illustrating a non-pythagorean coloring of the first 7824 integers. A white square can be either red or blue. Notice that 3 is blue and 5 is red, so the pythagorean triple $(3, 4, 5)$ is not uniformly colored. Similarly for (5, 12, 13), since 12 and 13 are blue. The triples (105, 608, 617) and (207, 224, 305) can also be checked in this corner. Here is the entire array, due to Marijn Heule and reproduced from Nature 534 17-18 with permission.

The proof involved checking each of the approximately $10^{2300}$ ways of coloring the first 7,825 integers, to see that there is always at least one uniformly colored Pythagorean triple. "The researchers took advantage of symmetries and several techniques from number theory to reduce the total number of possibilities that the computer had to check to just under 1 trillion. It took the team about 2 days running 800 processors in parallel on the University of Texas's Stampede supercomputer to zip through all the possibilities. The researchers then verified the proof using another computer program." As for whether it's math or not: "If mathematicians' work is understood to be a quest to increase human understanding of mathematics, rather than to accumulate an ever-larger collection of facts, a solution that rests on theory seems superior to a computer ticking off possibilities." Lamb contrasts the situation here with the status of the previous longest-proof record-holder: a "13-gigabyte proof from 2014, which solved a special case of a question called the Erdös discrepancy problem. A year later, mathematician Terence Tao of the University of California, Los Angeles, solved the general problem the old-fashioned way -- a much more satisfying resolution."

"Topology at the Tonys"

Evelyn Lamb, again. That's the title for her blog posting on the Scientific American website, June 14, 2016. Lamb reports that at the Tony Award presentations this year, the cast of the musical "Waitress," currently on Broadway, performed a number which ended with a move from the Philippine folk dance Binasuan: a glass of water held on the palm is rotated $720^{\circ}$ without spilling, with the arm coming back to its original position ("Waitress" uses pies). Where's the topology? As Lamb explains, the "pie trick" illustrates a property of curves in the space of rotations. The curve starting from the zero rotation and ending at $360^{\circ}$ gets your hand back to where it started but twists your arm; surprisingly, another traversal of the same curve, so going from $360^{\circ}$ to $720^{\circ}\!$, undoes the twist. A wonderful computer animation of the phenomenon has been posted by Jason Hise.

"L.A. Math"

Princeton University Press is bringing us "L.A. Math: Romance, Crime, and Mathematics in the City of Angels," by James D. Stein, a math professor emeritus at Cal State, Long Beach. It was reviewed by Matthew Riesz for Times Higher Education: "By the end, readers should have acquired some of the basics of algebra and geometry, probability, game theory and the mathematics of elections. On the way, Professor Stein has fun adopting the voice of a hard-boiled, wise-cracking private eye." And by Brian Clegg on his blog: "It has always seemed that it would be a great idea to write fiction which managed to painlessly get across ideas in science or mathematics, but usually the outcome of attempting to do this is something distinctly worthy that lacks any entertainment or effectiveness as a narrative. In L.A. Math, James Stein has managed the closest approximation to getting it just right I've yet to see. The stories work as detective tales, but the denouement relies not on sophisticated detection but on mathematical deduction." Full review.

Princeton has posted a cute movie-style trailer for the book.

"Don't call me a prodigy: the rising stars of European mathematics"

DW (Deutsche Welle, the German foreign news service) posted a bulletin with that title on July 23, 2016, from the European Congress of Mathematics. As their reporter Helena Kaschel tells us, "One main takeaway: some of the continent's leading mathematicians are surprisingly young--and very down-to-earth." Kaschel interviewed three of them.

  • Peter Scholze (Bonn; arithmetic algebraic geometry). "I don't believe you always have to understand everything in mathematics. Gerd Faltings ... regularly holds a lecture on arithmetic geometry at Bonn University. I used to go there as a student and I would never understand anything. But in hindsight I feel like I learned so much during that time. There's this misconception that certain parts of lectures are pointless if you don't get it straight away."
  • James Maynard (Oxford; number theory). "It's unfortunate that maths has this air of being inaccessible. Professional mathematicians quite often have the same experiences as normal people, that there's an abstract concept and they don't get it straight away. In some ways it can be quite damaging for mathematics that there's this idea of the lone genius."
  • Sara Zahedi (KTH; numerical analysis), who came to Sweden from Iran at age 10. "I didn't have any friends and I didn't know any Swedish. But math was a language I understood. In math class, I was able to communicate with my peers and I was able to make friends by solving problems with them." "We should be reaching out to much younger age groups and educating them about how math can be applied in real life. I also think we should teach programming in schools."


U.S. postage stamp for "Stand and Deliver"

As NBC News reported on July 17, 2016, the United States Postal Service has issued a stamp celebrating Jaime Escalante, the high-school calculus teacher immortalized in the movie "Stand and Deliver" (1988), "one of the most viewed movies in U.S. film history." The stamp shows Escalante in front of a blackboard with recognizable bits ($e^x, 2x~dx$) of calculus notation. Edward James Olmos, who received an Oscar nomination for his portrayal of Escalante in the film, was at the ceremony: "If it wasn't for teachers, none of us would be where we are today. God bless Jaime Escalante and God bless the United States Postal Service."

U.S. team wins I.M.O.

The story, by Gary Antonick, ran on the New York Times website as "U. S. Team Wins First Place at International Math Olympiad" (July 18, 2016). Antonick lists the team members: Ankan Bhattacharya, Michael Kural, Allen Liu, Junyao Peng, Ashwin Sah, and Yuan Yao; gives two of the problems, including the difficult IMO 2016 Problem 3:

  • Let $P = A_1 A_2 \dots A_k$ be a convex polygon on the plane. The vertices $A_1, A_2, \dots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$;
and has a leisurely discussion with the team's coach, Po-Shen Loh. To explain the importance of conceptualization in problem solving Loh uses a metaphor: a question, written in English and in Bengali. "When you look at the English vesion of this question ..., your brain is not memorizing where the squiggles are. Your brain has conceptualized it already, and then compresses the information. You have no problem dealing with it. So -- what's the difference between a top performer and everyone else? If you look at mathematics--if you've built the concept structure, when you reason about the problem, you're reasoning in large concepts. ... It's not a miracle. It's just about whether your brain has partitioned the concept map, and whether you can deal with entire concepts as primitive arguments as opposed to working with each little letter at a time."


The New Yorker and Andrew Hacker at MoMath

The "Talk of the Town" section of the June 27, 2016 New Yorker included a piece by their staff writer Rebecca Mead, whom they had dispatched downtown to cover a debate between Andrew Hacker, the iconoclastic scourge of the mathematical establishment, and James Tanton, "the mathematician at large of the Mathematical Association of America," staged on the premises of the National Museum of Mathematics, or MoMath. "Hacker outlined his case: mastery of the high-school-math sequence--algebra, geometry, calculus--is unnecessary for most students, and by making math a requirement for graduation and college entrance, the U.S. educational system sets up for failure millions whose talents lie elsewhere." As for the assertion that math sharpens the mind, Mead quotes him: "There is no evidence whatever that mastering mathematics makes you agile and adept in other fields." Mead reports that Hacker gives the "cautionary example of Paul Wolfowitz, the former Deputy Secretary of Defense and an architect of the Iraq War." Then Tanton takes the floor: "The issue is, How do we teach the subject? Do we teach with beauty and joy and humanness?" Mead, who confesses to "weeping with frustration over her fifth grader's math homework," is not convinced. Later, she asks Hacker whether he thinks "math really is harder than other classes that students are required to take. 'Unqualifiedly, yes,' he said. 'Every other subject is about something. Poetry is about something. Even most modern art is about something.' He looked around and lowered his voice to a whisper. 'Math is about nothing.'"

Tony Phillips
Stony Brook University
tony at