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There could thus be, at least in theory, a volume of Principia Mathematica devoted entirely to exploring the question of which integers are, and which are not, the sum of two squares. For instance, 41 is the sum of 16 and 25, and there are infinitely many other integers that can be made by summing two squares. Call them members of Class A. On the other hand, 43 is not the sum of any pair of squares, and likewise, there are infinitely many other integers that cannot be made by summing two squares. Call them members of Class B. (Which class is 109 in? What about 133?) Fully fathoming this elegant dichotomy of the set of all integers, though a most subtle task, had been accomplished by number theorists long before Gödel’s birth.

Analogously, one could imagine another volume of Principia Mathematica devoted entirely to exploring the question of which integers are, and which are not, the sum of two primes. For instance, 24 is the sum of 5 and 19, whereas 23 is not the sum of any pair of primes. Once again, we can call these two classes of integers “Class C” and “Class D”, respectively. Each class has infinitely many members. Fully fathoming this elegant dichotomy of the set of all integers represents a very deep and, as of today, still unsolved challenge for number theorists, though much progress has been made in the two-plus centuries since the problem was first posed.

Mixing Two Unlikely Ideas: Primes and Squares

Before we look into Gödel’s unexpected twist-based insight into PM, I need to comment first on the profound joy in discovering patterns, and next on the profound joy in understanding what lies behind patterns. It is mathematicians’ relentless search for why that in the end defines the nature of their discipline. One of my favorite facts in number theory will, I hope, allow me to illustrate this in a pleasing fashion.

Let us ask ourselves a simple enough question concerning prime numbers: Which primes are sums of two squares (41, for example), and which primes are not (43, for example)? In other words, let’s go back to Classes A and B, both of which are infinite, and ask which prime numbers lie in each of them. Is it possible that nearly all prime numbers are in one of these classes, and just a few in the other? Or is it about fifty–fifty? Are there infinitely many primes in each class? Given an arbitrary prime number p, is there a quick and simple test to determine which class p belongs to (without trying out all possible additions of two squares smaller than p)? Is there any kind of predictable pattern concerning how primes are distributed in these two classes, or is it just a jumbly chaos?

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