I Am a Strange Loop

What could that unknown prime divisor be? It can’t be 2, because 2 divides Q itself, which is just one step below Q + 1, and two even numbers are never located at a distance of 1 from each other. It also can’t be 3, because 3 likewise divides Q itself, and numbers divisible by 3 are never next-door neighbors! In fact, whatever prime p that we select from the Club, we find that p can’t divide Q + 1, because p divides its lower neighbor Q (and multiples of p are never next-door neighbors — they come along only once every p numbers). And so reasoning has shown us that none of the members of the Finite, Closed Club of Primes divides Q + 1.

But above, I observed (and I asked you to remember) that Q + 1, being composite, has to have a prime divisor. Sting! We have been caught in a trap, painted ourselves into a corner. We have concocted a crazy number — a number that on the one hand must be composite (i.e., has some smaller prime divisor) and yet on the other hand has no smaller prime divisor. This contradiction came out of our assumption that there was a Finite, Closed Club of Primes, gloriously crowned by P, and so we have no choice but to go back and erase that whole amusing, suspect vision.

There cannot be a “Great Last Prime in the Sky”; there cannot be a “Finite, Closed Club of All Primes”. These are fictions. The truth, as we have just demonstrated, is that the list of primes goes on without end. We will never, ever “fall off the Earth”, no matter how far out we go. Of that we now are assured by flawless reasoning, in a way that no finite amount of computational sailing among seas of numbers could ever have assured us.

If, perchance, coming to understand why there is no last prime (as opposed to merely knowing that it is the case) was a new experience to you, I hope you savored it as much as a piece of chocolate or of music. And just like such experiences, following this proof is a source of pleasure that one can come back to and dip into many times, finding it refreshing each new time. Moreover, this proof is a rich source of other proofs — Variations on a Theme by Euclid (though we will not explore them here).

The Mathematician’s Credo

We have just seen up close a lovely example of what I call the “Mathematician’s Credo”, which I will summarize as follows:

X is true because there is a proof of X;

X is true and so there is a proof of X.

Notice that this is a two-way street. The first half of the Credo asserts that proofs are guarantors of truth, and the second half asserts that where there is a regularity, there is a reason. Of course we ourselves may not uncover the hidden reason, but we firmly and unquestioningly believe that it exists and in principle could someday be found by someone.

To doubt either half of the Credo would be unthinkable to a mathematician. To doubt the first line would be to imagine that a proved statement could nonetheless be false, which would make a mockery of the notion of “proof ”, while to doubt the second line would be to imagine that within mathematics there could be perfect, exceptionless patterns that go on forever, yet that do so with no rhyme or reason. To mathematicians, this idea of flawless but reasonless structure makes no sense at all. In that regard, mathematicians are all cousins of Albert Einstein, who famously declared, “God does not play dice.” What Einstein meant is that nothing in nature happens without a cause, and for mathematicians, that there is always one unifying, underlying cause is an unshakable article of faith.

No Such Thing as an Infinite Coincidence

We now return to Class A versus Class B primes, because we had not quite reached our revelation, had not yet experienced that mystical frisson I spoke of. To refresh your memory, we had noticed that each line was characterized by differences of the form 4n — that is, 4, 8, 12, and so forth. We didn’t prove this fact, but we observed it often enough that we conjectured it.

The lower line in our display starts out with 3, so our conjecture would imply that all the other numbers in that line are gotten by adding various multiples of 4 to 3, and consequently, that every number in that line is of the form 4n + 3. Likewise (if we ignore the initial misfit of 2), the first number in the upper line is 5, so if our conjecture is true, then every subsequent number in that line is of the form 4n + 1.