The Clatter of the Primes

The trouble began, as trouble often does, with a rivalry between friends. It took place during the Big Before, when numbers and operations were new and still figuring themselves out, and none of them had any idea what a universe was or whether having one would be a good idea.

Plus said to Times “No offense, friend, but I’m just better at building numbers than you are. Starting from 1, the smallest number, I can build lots of new numbers: 1+1 is 2, 1+1+1 is 3, and so on. But look at you! 1×1 is just 1. 1×1×1? 1 again. And so on. Boring!”

Times naturally became defensive. “Now that’s just not fair. You’re using the wrong building block. Instead of 1, try 2.” And the number 2 began to twinkle. “2×2 is 4. 2×2×2 is 8. And so on. See, I get new numbers, just like you, and mine are bigger than yours!”

Plus said “I can get all those numbers, and more; it just takes me longer. But I get some numbers you can’t get. 3 is 1+1+1, but you’ll never get 3 by multiplying 2’s.”

Times, thinking quickly, retorted, “I never said I could get everything from 2’s. I also use 3 as a building block.” Then the number 3 began to twinkle. “For instance, with 2 and 3, I can get 2, 4, 8, and so on, and 3, 9, and so on. And I mix 2’s and 3’s, so I get 6 and lots of other numbers too.”

Plus said “What about 5? How do you get 5 by multiplying 2’s and 3’s?”

Times airily answered “Oh, I never said 2 and 3 would be enough! 5 is another one of my building blocks.” And the number 5 began to twinkle.

Plus asked “How many of these building blocks do you have?”

Times didn’t answer right away. As I said, numbers and operations were still figuring themselves out. But Times wasn’t going to retreat from this showdown with Plus; Times had to prove that Times, like Plus, had an orderly way of building up all the numbers.

PRIME NUMBERS

The building blocks that Times needed are called prime numbers. The ancient Greeks got interested in prime numbers and figured out that there are infinitely many of them, though they didn’t phrase it that way; they said something closer to “No matter how many primes you’ve found, there’s always a prime you haven’t found.”

Before we see why you can never run out of primes, let’s make a table that shows how the numbers 1 through 10 can be written as products of primes. I’ve left a blank in the first row, because 1 is a special case: it’s sometimes called the product of no primes at all, but if that doesn’t make sense to you, don’t worry about it.

One pattern in the table is that no prime occurs in two consecutive rows. You don’t see a 2 in two consecutive rows because when two counting numbers are consecutive, one of them is a multiple of 2 and the other one is not. Likewise you don’t see two consecutive counting numbers that are both multiples of 3, because the multiples of 3 are spaced three apart, like the downbeats in a waltz. The same goes for 5 and 7, and if we had a bigger table, it would apply to larger primes as well. The primes that are factors of the counting number n cannot be factors of the counting number n+1.

Putting it differently: From an additive-construction perspective, the numbers n and n+1 are very similar (just tack on an extra “+1” at the end of Plus’s way of the number n and you’ve got Plus’s way of writing n+1), but from a multiplicative-construction perspective, n and n+1 are as different as mosques and mosquitos (to pick two words that are adjacent in the dictionary but have nothing to do with each other). Their representations as products are as different as can be.

PRIMES WITHOUT END

Now I can explain why you’ll never run out of primes. Take all the primes you know, find a big number n that’s a multiple of all of them (say by multiplying them all together), and then add 1 to that big number. Your new, ever-so-slightly bigger number, n+1, can’t be a multiple of any of the primes you know (because n was a multiple of all of them), so it’s either a prime itself (one that you didn’t know) or it’s a product of two or more primes (also previously unknown to you). Either way, there are new primes to meet. (See Endnote #1.) So, for example, 2×3 + 1 is a new prime (hello 7) and 2×3×5 + 1 is a new prime (hello 31), and 2×3×5×7×11×13 + 1 is the product of two new primes (hello 59 and 509).

There’s a song about this proof: check out “Plentitudinous Primes” by Hannah Hoffman and Joel David Hamkins.

Notice that even though the proof tells you why there’s got to be another prime besides the ones you know (or maybe several mew primes) it doesn’t tell you what that new prime is (or what those several new primes are): n+1 could be prime, or it could be the non-prime product of two or more primes. The proof doesn’t tell you which. Mathematicians don’t really know to what extent one might expect the first situation to prevail as opposed to the second, though they’ve come up with some good guesses about special circumstances. For instance, say n itself is 2 times a prime (let’s write n = 2p). When 2p+1 is prime, we call p a Germain prime. Are there infinitely many Germain primes? Mathematicians think so, but they haven’t found a proof. Or, say n is a power of 2 (let’s write n = 2^k). When 2^k+1 is prime, we call it a Pierre Fermat prime. Are there infinitely many Pierre Fermat primes? Mathematicians think not, but they haven’t found a proof. (See Endnote #2.)

And that’s just one tiny corner of all the things we don’t know about primes, despite the efforts of a lot of hardworking and clever people. For instance, you may have heard about the twin prime conjecture, which concerns primes p for which p+2 is also prime (“twin primes”). Are there infinitely many twin primes? Mathematicians think so, but they haven’t found a proof.

MUSIC OR NOISE?

If you’re tired of my saying “Mathematicians think … but they haven’t found a proof”, then you probably won’t like number theory, which has a lot of we-don’t-knows. Number theory is the study of prime numbers. Well, not just prime numbers. It’s also the study of other things involving whole numbers, such as the question “When can an integer be written as the sum of two perfect squares?” but (surprise!) that question turns out to be a question about primes in disguise. And this phenomenon (hello again, prime numbers!) happens over and over when you ask questions about whole numbers.

Number theory is full of simple-sounding questions number theorists don’t have answers to, so if your archetype of mathematics is middle school arithmetic, where there’s a procedure to finding the answer to every question, or high school geometry, where every true proposition has a tidy proof, then number theory isn’t for you. In fact, if you suffer from trypophobia (an aversion to the sight of irregular patterns or clusters of small holes or bumps), then you should stop reading this essay right now, because “irregular patterns of bumps” is a pretty spot-on description of the primes.

Not everyone likes Stravinsky, either. His “Rite of Spring” caused a big to-do (though not a riot; see Endnote #3) in the spring of its premiere. Music-lovers are more used to dissonance nowadays than they were in 1913, but many people still find the piece unpleasant. Consider an eight-bar passage from near the beginning of the piece, about four minutes in. I can’t include an audio snippet here, but you can listen to a YouTube recording, such as the London Symphony Orchestra version, and advance to the four-minute mark if you’re in a hurry. Or you can listen to my rendition of this snippet using a synthetic piano (and an approximation to Stravinsky’s chord), if you’ve got a midi player:

http://mathenchant.org/082-rite.mid

Those accents on beats 10, 12, 18, 21, 25, and 30 don’t land when you expect them to, even when you remember that they don’t land when you expect them to (I’m guessing that this was the effect Stravinsky was aiming for). Sometimes the blows land two beats apart, sometimes three beats apart, sometimes more; we hear suggestions of order, only to have the patterns get broken.

The primes have some of that same unpredictability. Here I’ve rendered the numbers 1 through 720 using Stravinsky’s chord, where the primes are loud and the non-primes are soft.

http://mathenchant.org/082-primes.mid

If you listen closely, you’ll notice that every other beat (with one solitary exception) provides guaranteed safety against those accents, because all but one of the primes are odd. If you pretend you’re in fin-de-siecla Vienna and you listen to the MIDI file with your waltz-ears on, you’ll notice that (with one solitary exception) every third beat also provides safety, because only one of the primes is a multiple of three. And if you’re comfortable parsing music into groups of five beats or seven beats, you can hear a similar pattern of “safe” beats. (Alternatively, you can listen to the MIDI file

http://mathenchant.org/082-nonprimes.mid

in which I’ve accented the non-primes instead of the primes; now there’ll be certain beats on which you’re guaranteed to hear an accent, instead of guaranteed not to hear one.) But other than that, it’s hard for the musical mind to make sense of what it’s hearing. When rendered as a purely rhythmic composition, the primes form a music whose brief passages of order and pattern are only there to throw its savage randomness into sharper relief.

Except … the primes aren’t really random! Quite the opposite. Every individual number is either prime or isn’t; the primes are exactly where they inevitably must be. The irregularity of the primes and our attendant discomfort aren’t the result of choices made by a Stravinsky-ish Creator; the irregularity is an inexorable consequence of the laws of arithmetic, while the discomfort is the result of human nature and of our choice to look at (or listen to) the primes.

We can choose to look away; some people do. When Fermat tried to entice others into joining him on his forays into number theory, only a few of them took the bait. “We have no lack of better things to think about,” one of his pen-pals wrote. Even today some mathematicians react that way to the primes. And that’s fine! Once you get past the basics, mathematics branches into dozens of different subdisciplines, and there are enough different kinds of math to like that you can dislike two or even three of them and still be a respectable mathematician.

The study of primes is rife with frustration. Of course, frustration can be a wonderful thing if it’s inflicted not on you but on people who are trying to break into your house or hack into your computer accounts; the thorniness of primes turns out to be helpful when we design computer systems to thwart the efforts of bad actors. But that’s probably not news to you, if you’ve ever delved into the math that underlies cryptocurrency.

LIOUVILLE AND CHOWLA

But I do have actual news for you. Or at least, I can report on a discovery made less than a year ago by researchers Harald Helfgott and Maksym Radziwiłł. Their work concerns a question raised by the mathematician Sarvadaman Chowla back in 1965. Like all twentieth-century number theorists, Chowla was aware of the intriguing twin prime conjecture and equally aware that very little progress had been made toward resolving it. So he applied an old strategy of mathematical researchers: when faced with a problem you can’t solve, explore a related problem. Instead of looking at numbers that are 2 apart, as in the twin primes conjecture, Chowla looked at numbers that are 1 apart. Clearly they’re not both primes (leaving aside the case of 2 and 3), since one of them is even and 2 is the only even prime. Chowla’s question is about something called parity, or more precisely Liouville parity, named after the number theorist Joseph Liouville who first studied it. The Liouville parity of n has to do with how many primes you multiply together to get n. For instance, we might say that 12 is “Liouville-odd” because we write it as 2×2×3, a product of three primes (it’s okay to repeat a prime but you have to count the repeats).

Number theorists don’t actually use the terms “Liouville-even” and “Liouville-odd” the way I did; instead, they write “L(n) = +1″ and “L(n) = −1″ according to whether the number n is the product of an even number of primes or an odd number of primes. The Liouville function of n, denoted by L(n), is defined as −1 to the power of the number of factors when n is written as a product of primes. (It’s convenient to regard 1 as the product of no primes at all, which is to say, zero primes, so that L(1) = (−1)⁰ = 1.)

It’s known that in the long run, half of the positive integers are Liouville-even and half are Liouville-odd. For instance, from 1 to 1000 there are 493 Liouville-even integers and 507 Liouville-odd integers, while from 1 to 1,000,000 there are 499,735 Liouville-even integers, and 500,265 Liouville-odd integers. Even as the absolute error grows (from ±7 to ±265), the relative error shrinks (from ±1.4% to ±0.1%), getting ever-closer to 0%.

Here’s what you get if on the nth beat you play a C if n is Liouville-even and D if n is Liouville-odd.

http://mathenchant.org/082-liouville.mid

I don’t hear any long-term patterns; do you?

In contrast, here’s what you get when you play a random sequence of C’s and D’s determined by tossing a coin (or rather by using a commercial pseudorandom number generator that does a decent job of simulating a fair coin):

http://mathenchant.org/082-random.mid

Does it sound the same to you? Or can your musical ear discern a difference between the noise of random numbers and the music of Liouville’s function?

Chowla’s question concerned the relationship, if any, between the Liouville-parity of nearby numbers. For instance we might compare the Liouville-parities of n and n+1. If the coin-toss analogy holds, we might expect that, among the first 1000 positive integers, each of the four possibilities (n is L-odd and n+1 is L-odd; n is L-odd and n+1 is L-even; n is L-even and n+1 is L-odd; n is L-even and n+1 is L-even) will occur about 250 times. In fact, these four outcomes occur 261 times, 246 times, 247 times, and 246 times, respectively. Not a bad fit. And if Chowla’s guess is right, we can expect the fit to get better and better (proportionately) when we replace 1000 by ever-larger cutoffs.

NUMBERS AND GRAPHS

To find out how Helfgott and Radziwiłł improved on earlier results, check out Jordana Cepelewicz’s excellent article published in Quanta late last year. One of the main innovations of the new paper of Helfgott and Radziwiłł is the use of methods from graph theory to approach a problem from number theory. Number theory and graph theory were the two favorite subject areas of Paul Erdős; he would have loved to see them in such an intimate embrace.

Graph theory is often used to model social networks, and in a way, the math of Helfgott and Radziwiłł embodies a kind of social view of the positive integers, in which numbers “conspire” (or don’t conspire) to have related values of Liouville’s function.

If you know a little bit of applied graph theory, you may have heard of small-world networks, in which it’s possible to find a short chain of connections between any two nodes (think “six degrees of separation“). The work of Helfgott and Radziwiłł is based on a different way of measuring how well the connections bind the whole network together. The key word here is “expander“. I won’t give a technical definition, but I’ll give two examples of networks that fail to be expanders and one that does a decent job.

In each of our networks, the nodes being connected are the positive integers 1, 2, 3, … but the patterns of connections are different. In the first network, we create a one-way connection from i to j whenever j=i+1 or j=i+2. Thus, 1 gets connected to 2 and 3, 2 gets connected to 3 and 4, 3 gets connected to 4 and 5, etc.

Starting from 1, there are four journeys of length two we can take (1 to 2 to 3, 1 to 2 to 4, 1 to 3 to 4, and 1 to 3 to 5), leading us to three possible destinations: 3, 4, and 5. Starting from 1, there are only four possible destinations: 4, 5, 6, and 7. The number of destinations we can reach in k steps is always equal to k+1. So as k increases, the number of destinations reachable in k steps gets bigger, but it does not increase quickly.

What if instead we create a one-way connection from i to j whenever j=2i or j=3i?

Once again, the number of destinations reachable in k steps, starting from 1, is always equal to k+1 — a linear function of k.

But now, let’s mix addition and multiplication. What if we create a one-way connection from i to j whenever j=2i or j=2i+1?

Then starting from 1 we can take a one-step journey to 2 or 3, a two-step journey to 4, 5, 6, or 7, a three-step journey to 8, 9, 10, 11, 12, 13, 14, or 15, etc. The number of destinations now grows exponentially as a function of k. That’s the kind of expansion that expander graphs have.

You’ll have to read Cepelewicz’s article if you want to know more, and I hope you will. The paper of Helfgott and Radziwiłł doesn’t settle Chowla’s original problem, but it gives stronger results than anyone has gotten before, and raises hopes that the new approach will yield other fruits as well.

NUMBERS AND ORBITS

Meanwhile, other researchers are approaching Chowla’s conjecture from a totally different direction, using another branch of mathematics that came of age in the 20th century: ergodic theory. Vitaly Bergelson, El Houcein El Abdalaoui, Joanna Kulaga-Przymus, Mariusz Lemanczyk, Redmond McNamara, Florian Richter, Thierry de la Rue, and others have been applying ideas that originally arose in Henri Poincaré’s work on celestial mechanics. I learned about this work quite recently in a talk given by Richter and I won’t attempt to even sketch it here. But I think the way different communities of researchers are using very different tools to study the weird ways of primes reveals something about the fundamental unity of mathematics. When one is tackling a truly deep and obdurate phenomenon like the statistical regularity that lies hidden in the primes, all of humanity’s best mathematical tricks have a role to play. I believe this tells us how deep mathematics is, and how limited the human mind is when it comes to plumbing those depths. After all, our brains didn’t evolve to do math; when you’re as ill-suited to abstract mathematics as humans are, every little bit helps!

If you ask me to guess which community will prove Chowla’s conjecture, I think that’s the wrong question. A better question is, Which community will prove Chowla’s conjecture first?, because it’s likely that multiple approaches will ultimately pay off. But even that question bothers me, because the first proof isn’t always the one that’s easiest to read or most illuminating. A better question might be, Which community will prove Chowla conjecture best? But I don’t like that question either. Why must math research be a contest? Mathematics is the richer for having multiple paths to its truths. For instance, take the Prime Number Theorem, one of the cornerstones of analytic number theory. The ergodic theory folks have found a lovely new way to prove it. But I wouldn’t want this proof to displace earlier proofs. 

And now that I stop to think a bit harder about recent mathematical history, I think it’s quite possible that the first proof of Chowla’s conjecture will use ideas from graph theory and ideas from dynamics, and maybe some other areas of mathematics outside of number theory. Consider the work of Ben Green and Terence Tao on the existence of arithmetic progressions in the primes; it used ergodic theory, combinatorics (of which graph theory is a part), geometry, and harmonic analysis.

THE LIMITS OF KNOWLEDGE

There are many ways to understand Kurt Gödel‘s famous First Incompleteness Theorem (and even more ways to misunderstand it). One approach, popularized by computer scientist Douglas Hofstadter in his book “Gödel, Escher Bach,” is to view it as an illustration of the pitfalls of self-reference in formal systems. But you can also view the incompleteness theorem as illustrating how richly addition and multiplication interact.

If you create a mutilated version of number theory that includes addition but not multiplication (Presberger arithmetic), or one that includes multiplication but not addition (Skolem arithmetic), then you get a tidy mathematical universe in which all the questions you can ask can be answered without any need for creativity. It’s a lot like Leibniz’s rosy vision of a world in which questions of morality or public policy could be reduced to calculation. But if you include both addition and multiplication, then you’re in the realm of Peano arithmetic, and Gödel’s clever construction (based in part on the prime numbers) shows that you can generate assertions that are true (assuming that Peano arithmetic is consistent) but cannot be derived from the Peano axioms. Leibniz’s dream is doomed to failure, even within the precincts of mathematics.

Given the role that the primes played in toppling Leibniz’s dream, it’s not too paranoid to suspect that some properties of the prime numbers may elude human reason forever. The philosopher Timothy Morton coined the term “hyperobject” to denote a thing that is too large for the human mind to grasp. He had things like global warming in mind, but I think the set of prime numbers also qualifies.

EPILOGUE

Plus and Times looked up at the numbers that Times had set twinkling: 2, 3, 5, 7, and infinitely many others. And they saw that they were messy.

“Not as neat as the sequence of odd numbers,” said Plus.

“Not as tidy as the sequence of powers of two,” said Times.

They were quiet for a bit.

“Not as sweet as the sequence of perfect squares,” said Plus.

“Right,” said Times. “With squares, the spacing gets bigger and bigger as you go out. These numbers only get mostly farther apart as you go, and even then, only sort of.”

And they were quiet again for a while.

“Can’t say I like them much,” said Plus. “Bit of a jumble. Aren’t we arithmetic operations supposed to be about order, pattern, and regularity?”

“I don’t really like them either,” admitted Times. “I kind of regret making them now.”

“Yeah, well, but I made you make them. With my boasting, I mean,” said Plus, who then paused, as if the very next sentence would be hard to say. “But … it’s the strangest thing, but I kind of feel I ought to like them.”

“I feel the same way!” exclaimed Times. “Something inside me says I ought to try to learn to like them. I don’t want to be the sort of being who only likes things that are easy to like.”

Another long silence followed.

“Well,” said Plus to Times, “we certainly created a big mess when we came up with those primes! But I’m guessing that if we put our heads together we can figure them out.”

“We do work well together,” said Times to Plus shyly. 

And the two of them rose up together, still talking, into a firmament bright with ineffable yet inevitable constellations.

Thanks to Jeremy Cote, Noam Elkies, David Feldman, Rebecca Gans, J. Ruth Gendler, Sandi Gubin, Evelyn Lamb, and Evan Romer.

ENDNOTES

#1: This argument is often presented as a proof by contradiction, along the lines of:

“We want to prove that the set of primes is infinite. Let’s suppose it’s finite and see where that supposition leads. If there are only finitely many primes, we can multiply them all together to get a number n that’s divisible by all the primes. But then n+1 is a number that’s divisible by none of the primes, and that’s a contradiction because every counting number can be written as a product of primes. So there must be infinitely many primes after all.” 

This is the rhetorical gambit of reductio ad absurdum, in which one undermines a premise by showing that it leads to unacceptable conclusions. The resemblance to legal argumentation brings to mind the fact that Fermat himself was a jurist.

I recently learned about a wacky Delaware court case, “Joseph Alfred v. Walt Disney”, featuring a complainant who, in suing Disney for a purported breach of contract involving the idea of a flying car, somehow managed to invoke Euclid’s proof of the infinitude of the primes (as well as Star Trek, Star Wars, Game of Thrones, the epic of Gilgamesh, and other cultural touchstones). Judge Sam Glasscock archly summarized the complaint thus: “It is well-written and compelling. In fact, it can be faulted only for a single — but significant — shortcoming: it fails to state a claim on which relief could be granted. Therefore, I grant the Defendants’ Motion to Dismiss.” 

But I find an additional shortcoming in the section of the complaint that mentions Euclid. Here’s what the complainant wrote:

“The Walt Disney Corporation created an implied contract with the plaintiff when it changed its own policy against submitting unsolicited submissions by a third party. The plaintiff can infer an implied promise based on circumstances that exist in the ordinary course of dealing and common understanding. Why even take the teleconference call on July 22, 2014 if there were not mutual agreement that the campaign would be successful for the Disney Corporation? There is an often used mathematical principle to solve difficult theorems: to prove something, disprove the opposite (See Euclid’s proof on the infinity of prime numbers).”

But Euclid’s proof as Euclid wrote it was not a proof by contradiction. All he showed is that if you have three primes you can always find a fourth, and left the reader to apply the same idea to infer the more general fact that prime numbers are more than any assigned multitude.

#2. It’s time I confessed to some authorial mischief, specifically, a nomenclatural switcheroo, when I used the terms “Germain prime” and “Pierre Fermat prime”.

People usually call primes p for which 2p+1 is also prime “Sophie Germain primes” (as opposed to “Germain primes”), presumably to highlight the fact that Germain was a woman. Germain kept her gender secret during her studies, since at that time women in France weren’t allowed to attend universities. When Gauss, the pre-eminent mathematician of that age and a correspondent of hers, learned that the venerable “Monsieur LeBlanc” was in fact a woman, he was all the more impressed, and praised her for having “the noblest courage, extraordinary talent, and superior genius”.

On the other hand, people usually call primes of the form 2^k+1 “Fermat primes”, not “Pierre Fermat primes”.

Before we discuss the fact that mathematicians usually give Germain’s first name but not Fermat’s when talking about the classes of primes that bear their names, I should mention that Fermat’s story is linked to Germain’s in a deep way. Fermat is famous for his assertion that if n is a whole number bigger than 2, the equation xⁿ + yⁿ = zⁿ has no solutions in positive whole numbers. Germain was the first to outline and undertake a systematic approach to solving the problem not just for specific values of n but (she hoped) for all values of n > 3. She came close to proving that Fermat’s guess was correct whenever n is a Germain prime. Inasmuch as there appear to be infinitely many Germain primes, this work was a huge step beyond the one-exponent-at-a-time results of her predecessors. For more on this work, see “Voici ce que j’ai trouve”: Sophie Germain’s grand plan to prove Fermat’s Last Theorem” by Reinhard Laubenbacher and David Pengelley.

Nor were Germain’s mathematical talents limited to number theory. She won a prize for presenting the first mathematical account of what’s going on with Chladni plates.

So why does the great mathematician Fermat get referred to by his surname only while the great mathematician Germain gets referred to by her full name? Probably for the same reason that the great novelist Austen is often called “Jane Austen”: the default mathematician, like the default novelist, is seen as male, so exceptions to the stereotype get marked.

My question is, does including Germain’s first name help or hinder the cause of equality for women? In an ideal world, free of gender prejudice, it would of course be pernicious to introduce different nomenclatural standards for men and women. But we don’t live in such a world, and until we do, maybe it’s good to remind ourselves and others that some great mathematicians were women.

Then again, think back to what went on in your mind (please be honest, if only with yourself) when I called them “Germain” and “Pierre Fermat”. Giving someone a first name brings them down to human scale; conversely, omitting that name turns them into someone severe, almost inhuman, chiseled out of granite. To put it starkly: “Prof. Germaine” sounds like a colleague whose opinion you would defer to at department meetings, whereas “Sophie” sounds like a colleague whom you might ask to refill the department coffee maker. 

When we selectively humanize women and not men in some profession, we diminish women collectively. Specifically, we make women seem warmer but less competent. Here I am leaning on ideas from social psychology that suggest that, although warmth and competence are compatible, we tend to see them as mutually exclusive, and to the extent that we see evidence that a person possesses one of the two traits, we tend to infer that they lack the other.

So what do you think?  “Germain primes”, or “Sophie Germain primes”? And more broadly, what’s the best way to disable the gendered stereotype of mathematicians that we all tend to have regardless of our gender? Please share your open-minded, curious and polite thoughts in the Comments!

Oh, and getting back to Pierre, you may note that I called him “Pierre Fermat” not “Pierre de Fermat”, as is common. He earned the “de” when he become a government official in Toulouse, which I’m sure was very nice for him, but what does his title have to do with his mathematics?

#3: I have to confess that I uncritically accepted the standard legend of the opening night riot until Evelyn Lamb sent me a link to Linda Shaver-Gleason’s essay “Did Stravinsky’s The Rite of Spring incite a riot at its premiere?” I found the essay illuminating in many ways. Not only did it debunk a myth I’d believed for half a century, but it also situated “The Rite of Spring” in a broader cultural context. The most eye-opening part of the essay for me was a video showing a 1987 recreation of the original choreography: it made me realize that the jerky movements of the dancers could be construed as comical. I’d always thought of the piece as Serious And Important Art, but seeing those men in conical hats thrashing like oversized muppets made me understand the reaction of the aristocrats in the balcony (the ones whose laughter angered the music lovers, thereby setting in motion a disturbance that legend magnified into a riot). It’s a commonplace that a major ingredient of humor is surprise; that’s one of the things that makes the wrong notes in a P.D.Q. Bach piece funny. So it stands to reason that accents that arrive when you don’t expect them to could be funny as well.

And this led me back to thinking about the primes. Erdős once said “God may not play dice with the universe, but something strange is going on with the prime numbers,” and he is not the only mathematician to have thought so. Might mathematical culture someday reach a vantage point of sophistication from which primes are seen, not as funny-strange, but as funny-ha-ha?