I Am a Strange Loop

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.

This activity is called, as you well know, “finding a proof ”, or stated otherwise, turning a conjecture into a theorem. The late great eccentric Hungarian mathematician Paul Erdös once made the droll remark that “a mathematician is a device for turning coffee into theorems”, and although there is surely truth in his witticism, it would be more accurate to say that mathematicians are devices for finding conjectures and turning them into theorems.

What underlies the mathematical mindset is an unshakable belief that whenever some mathematical statement X is true, then X has a proof, and vice versa. Indeed, to the mathematical mind, “having a proof ” is no more and no less than what “being true” means! Symmetrically, “being false” means “having no proof ”. One can find hints of a perfect, infinite pattern by doing numerical explorations, as we did above, but how can one know for sure that a suspected regularity will continue forever, without end? How can one know, for instance, that there are infinitely many prime numbers? How do we know there will not, at some point, be a last one — the Great Last Prime P?

If it existed, P would be a truly important and interesting number, but if you look at a long list of consecutive primes (the list above of primes up to 100 gives the flavor), you will see that although their rhythm is a bit “bumpy”, with odd gaps here and there, the interprime gaps are always quite small compared to the size of the primes involved. Given this very clear trend, if the primes were to run out all of a sudden, it would almost feel like falling off the edge of the Earth without any warning. It would be a huge shock. Still, how do we know this won’t happen? Or do we know it? Finding, with the help of a computer, that new primes keep on showing up way out into the billions and the trillions is great, but it won’t guarantee in rock-solid fashion that they won’t just stop all of a sudden somewhere out further. We have to rely on reasoning to get us there, because although finite amounts of evidence can be strongly suggestive, they just don’t cut the mustard, because infinity is very different from any finite number.

Sailing the Ocean of Primes and Falling off the Edge

You probably have seen Euclid’s proof of the infinitude of the primes somewhere, but if not, you have missed out on one of the most crucial pillars of human knowledge that ever have been found. It would be a gap in your experience of life as sad as never having tasted chocolate or never having heard a piece of music. I can’t tolerate such crucial gaps in my readers’ knowledge, so here goes nothing!

Let’s suppose that P, the Great Last Prime in the Sky, does exist, and see what that supposition leads to. For P to exist means that there is a Finite, Closed Club of All Primes, of which P itself is the glorious, crowning, final member. Well then, let’s boldly multiply all the primes in the Closed Club together to make a delightfully huge number called Q. This number Q is thus divisible by 2 and also by 3, 5, 7, 11, and so forth. By its definition, Q is divisible by every prime in the Club, which means by every prime in the universe! And now, for a joyous last touch, as in birthday parties, let’s add one candle to grow on, to make Q + 1. So here’s a colossal number that, we are assured, is not prime, since P (which is obviously dwarfed by Q) is the Great Last Prime, the biggest prime of all. All numbers beyond P are, by our initial supposition, composite. Therefore Q + 1, being way beyond P and hence composite, has to have some prime divisor. (Remember this, please.)