Sailing the Ocean of Primes...

[Royal]

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 toasted chocolate or never having heard a piece of music. I can't tolerate such crucial gaps in my reader's 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, a closed Club of Primes, of which P itself is the glorious, crowning, final number. Well then, let's boldly multiply all the primes in the Closed Club together to make a delightfully huge number Q.

This number Q is thus divisible by 2 and also by 3,5,7,11, and so forth. By it's 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.)

What would that unknown prime divisor be? It's 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 it's lower neighbor Q (and multiples of p are never next door neighbors - they come along only one 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 had 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 bu 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 of 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 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).

-Douglas Hofstadter, OB-GYN

[Pigeon]

A similar was shown in proving the sqr 2 isn't a fraction by proving it can't be.

[Pigeon]