Sometimes in math we’ll understand one aspect of a problem very well, while at the same time we understand another aspect of a problem very poorly. For instance, take the prime numbers. According to the prime number theorem, the number of prime numbers below x is approximately given by:
Where pi(x) is the prime counting function and ln(x) is the natural logarithm. As you keep counting your way up the number line, you’ll encounter more and more primes. They thin out and become more and more rare, but nonetheless there’s an infinite number of them as you keep going. The number that you encounter is roughly x/ln(x), and that approximation gets better and better as a percentage as you go up.
Following from this, the density of prime numbers near x will be about 1/ln(x) and the nth prine number will be approximately n*ln(n). Let’s take a look at a quick example. If x = 1,000,000, we should expect that there are about 1,000,000/ln(1,000,000) = 72,382 prime numbers below a million. In fact there are 78,498. Not too bad.
We’d also expect the millionth prime number to have a magnitude of roughly 1,000,000 * ln(1,000,000) = 13,815,501. It’s actually 15,485,863, which is within a few percent error.
Now that’s just the tip of the iceberg in terms of what we know about the prime numbers. Literally millions of pages have been written about the prime numbers and their relationship with everything from complex analysis to group theory to encryption.
But other things are a blank mystery. Consider pairs of prime numbers, where you have a number p and its twin p + 2 and both are prime. These are the “twin primes” and the first few are (3, 5), (5, 7), (11, 13), and (17, 19).
The function that counts them in analogy to the prime number theorem above is our Sunday Function. I’d write it down for you except I have no idea what it is. Neither does anyone else. We don’t even know if there’s an infinite number of twin primes, for all we know the “twin prime counting function” might flatline somewhere out along the number line because there just aren’t any more of them. Who knows?
Now assuming some statistical properties of the prime numbers, it’s possible to make educated guesses about the distribution of the twin primes. Guesses in math don’t count for much no matter how plausible they are, but it’s at least a place to start. And by plausibly guessing, mathematicians have showed some good reasons to think that the twin prime counting function behaves roughly as x / (ln(x))^2. It’s provably not any bigger than that in the sense of O notation, anyway. But that certainly doesn’t exclude an O(1) (in the notation) flatline because there’s only a finite number of twin primes.
If you can actually figure out what the Sunday Function is explicitly, you’ll be a math celebrity and you just might get yourself a Fields Medal (there’s no Nobel for math for some reason). Give it a try!