In which Carrie appreciates beauty and excellence in mathematical language…
Oh, man, this is so cool! Let me tell you about one small learning I had yesterday in the setting of the Metolius River Nature Reserve with a friend of mine. Let’s call him the Mystic, since I’m mystified by math.
Mystic asks: Do you believe prime numbers are finite or infinite?
Me: Finite. (It is what I learned in school, after all, and really it doesn’t make intuitive sense for there to be an endless amount of these…)
Mystic: Would you like to be proven wrong?
Me: Oh, yes! (I do love any attempt at this, especially from one boasting of such confidence.)
So he proceeds…
Imagine the finite list of prime numbers. If I had enough time I’d write them all down on a piece of paper. Call the last prime number “P”.
Now, let’s multiply all of the numbers on this list together. It will take a long time, but imagine we have done this and come up with our result. We’ll call this result “C” for composite. This C is the product of multiplying all the primes in existence from prime number 2 to prime number P.
My mystic then begins to demonstrate the basic principles of prime numbers for a reminder on how they work. This tutelage delights and reminds me of when Socrates demonstrates that learning is possible (with also the metaphysical implication that all knowledge is remembered knowledge) by “teaching”Meno’s house slave geometry. So, with his stick in hand, my mystic draws in the dirt numbers 1 through 9.
He then asks me: “What are the divisors of (C + 1)?” It cannot be divisible by 2, because C is divisible by 2, so the next even number would be C + 2. Nor can it be divisible by 3, because C is divisible by 3, so the next multiple of 3 would be C + 3. The same argument holds for every prime on the list, so (C + 1) is not divisible by any prime. That makes it a prime larger than P, which contradicts the hypothesis.
Therefore, there can’t be a largest prime number.
For someone who isn’t good at math but curious all the same, this Socratic proof in the dirt, near a rollicking river, surrounded by Pileated woodpecker excavated snags, and in the company of a wonderful person who makes magic, was only a infinitesimally small part of the beauty of the day.
(Special thanks to Silas for his clarification of Euclid’s Theorem.)