Douglas Hofstadter - I Am a Strange Loop

To some readers, these may seem like peculiar or even unnatural questions to tackle, but mathematicians are constitutionally very curious people, and it happens that they are often deeply attracted by the idea of exploring interactions between concepts that do not, a priori, seem related at all (such as the primes and the squares). What often happens is that some kind of unexpected yet intimate connection turns up — some kind of crazy hidden regularity that feels magical, the discovery or the revelation of which may even send mystical frissons up and down one’s spine. I, for one, shamelessly admit to being highly susceptible to such spine-tingling mixtures of awe, beauty, mystery, and surprise.

To get a feel for this kind of thing, let us take the list of all the primes up to 100 — 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 — a rather jumbly, chaotic list, by the way — and redisplay it, highlighting those primes that are sums of two squares (that is, Class A primes), and leaving untouched those that are not (Class B primes). Here is what we get:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,…

Do you see anything interesting going on here? Well, for one thing, isn’t it already quite a surprise that it seems to be a fairly even competition? Why should that be the case? Why shouldn’t either Class A or Class B be dominant? Will either the Class A primes or the Class B primes take over after a while, or will their roughly even balance continue forever? As we go out further and further towards infinity, will the balance tend closer and closer to being exactly fifty–fifty? If so, why would such an amazing, delicate balance hold? To me, there is something enormously alluring here, and so I encourage you to look at this display for a little while — a few minutes, say — and try to find any patterns in it, before going on.

Pattern-hunting

All right, reader, here we are, back together again, hopefully after a bit of pattern-searching on your part. Most likely you noticed that our act of highlighting seems, not by intention but by chance (or is it chance?), to have broken the list into singletons and pairs. A hidden connection revealed?

Let’s look into this some more. The boldface pairs are 13–17, 37–41, and 89–97, while the non-boldface pairs are 7–11, 19–23, 43–47, 67–71, and 79–83. Suppose, then, that we replace each pair by the letter “P” and each singleton by the letter “S”, retaining the highlighting that distinguishes Class A from Class B. We then get the following sequence of letters:

S, S, S, P, P, P, S, S, P, P, S, S, S, P, S, P, P, . . .

Is there some kind of pattern here, or is there none? What do you think? If we pull out just the Class-A letters, we get this: SSPSPSSSP; and if we pull out just the Class-B letters, we get this: SPPSPSPP. If there is any kind of periodicity or subtler type of rhythmicity here, it’s certainly elusive. No simple predictable pattern jumps out either in boldface or in non-boldface, nor did any jump out when they were mixed together. We have picked up a hint of a quite even balance between the two classes, yet we lack any hints as to why that might be. This is provocative but frustrating.

People who Pursue Patterns with Perseverance

At this juncture, I feel compelled to point out a distinction not between two classes of numbers, but between two classes of people. There are those who will immediately be drawn to the idea of pattern-seeking, and there are those who will find it of no appeal, perhaps even distasteful. The former are, in essence, those who are mathematically inclined, and the latter are those who are not. Mathematicians are people who at their deepest core are drawn on — indeed, are easily seduced — by the urge to find patterns where initially there would seem to be none. The passionate quest after order in an apparent disorder is what lights their fires and fires their souls. I hope you are among this class of people, dear reader, but even if you are not, please do bear with me for a moment.

It may seem that we have already divined a pattern of sorts — namely, that we will forever encounter just singletons and pairs. Even if we can’t quite say how the S’s and P’s will be interspersed, it appears at least that the imposition of the curious dichotomy “sums-of-two-squares vs. not-sums-of-two-squares” onto the sequence of the prime numbers breaks it up into singletons and pairs, which is already quite a fantastic discovery! Who would have guessed?

Unfortunately, I must now confess that I have misled you. If we simply throw the very next prime, which is 101, into our list, it sabotages the seeming order we’ve found. After all, the prime number 101, being the sum of the two squares 1 and 100, and thus belonging to Class A, has to be written in boldface, and so our alleged boldface pair 89–97 turns out to be a boldface triplet instead. And thus our hopeful notion of a sequence of just S’s and P’s goes down the drain.

What does a pattern-seeker do at this point — give up? Of course not! After a setback, a flexible pattern-seeker merely regroups. Indeed, taking our cue from the word just given, let us try regrouping our sequence of primes in a different fashion. Suppose we segregate the two classes, displaying them on separate lines. This will give us the following:

Yes square + square: 2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101,…

No square + square: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83,…

Do you see anything yet? If not, let me give you a hint. What if you simply take the differences between adjacent numbers in each line? Try it yourself — or else, if you’re very lazy, then just read on.

In the upper line, you will get 3, 8, 4, 12, 8, 4, 12, 8, 12, 16, 8, 4, whereas in the lower line you will get 4, 4, 8, 4, 8, 12, 4, 12, 8, 4, 8, 4. There is something that surely should jump out at even the most indifferent reader at this point: not only is there a preponderance of just a few integers (4, 8, and 12), but moreover, all these integers are multiples of 4. This seems too much to be merely coincidental.

And the only larger number in these two lists — 16 — is also a multiple of 4. Will this new pattern — multiples of 4 exclusively — hold up forever? (Of course, there is that party-pooper of a ‘3’ at the very outset, but we can chalk it up to the fact that 2 is the only even prime. No big deal.)

Where There’s Pattern, There’s Reason

The key thought in the preceding few lines is the article of faith that this pattern cannot merely be a coincidence. A mathematician who finds a pattern of this sort will instinctively ask, “Why? What is the reason behind this order?” Not only will all mathematicians wonder what the reason is, but even more importantly, they will all implicitly believe that whether or not anyone ever finds the reason, there must be a reason for it. Nothing happens “by accident” in the world of mathematics. The existence of a perfect pattern, a regularity that goes on forever, reveals — just as smoke reveals a fire — that something is going on behind the scenes. Mathematicians consider it a sacred goal to seek that thing, uncover it, and bring it out into the open.