Numbers are funny things. Perhaps one of the funniest things about numbers is the following: for every number, there are infinitely many numbers larger than that number. For example, there are infinitely many numbers larger than 17, and there are also infinitely many numbers larger than
17,235,503,352,023,325,214,032,491,412,982,571.
Now, to us humans, one of the two numbers that I just mentioned seems way, way bigger than the other. And that's true, in some sense.
Obviously, if I asked you which of those two numbers you'd prefer to have in dollar bills, you'd probably immediately choose one over the other (but when you make that decision, you should really only spend a tiny fraction of those dollar bills, given the inflationary affects that would ensue (not to mention the storage issues – try estimating how much real estate you'd need just to house that many dollar bills)).
But suppose I proved a statement that held true "up to" one of those numbers (namely, a pattern about numbers that holds up to 17, and a pattern that holds up to that other number – 17,235,503,...). For what percentage of numbers have I proven my statement?
The answer, in both cases, is 0%. That's because of three facts: 1) in both cases I've only proven my statement for finitely many numbers, 2) there are infinitely many numbers, and 3) any finite number divided by an infinitely large number is 0.
"Almost True" Statements
While all these statements are probably familiar to some extent, they're incredibly hard to wrap one's head around and fully internalize. Showing that a statement is true for the first billion quintillion whole numbers is certainly more promising than only showing it for the first hundred whole numbers.
But to the gods of numbers, a billion quintillion is still not even a single grain of sand on the infinite shores of numbers. There's no reason why some subtle patter in numbers can't take hold only after, say, a million billion quintillion.
If there were some mathematical statement about numbers that held true up to a billion quintillion and then failed thereafter, we would likely refer to that statement as "almost true," because to our minds a number like a billion quintillion "feels like" infinity. But the following trivial example will hopefully help us think differently.
Claim: All numbers are less than a billion quintillion.
Numerical Evidence: We have checked that this claim holds true for all numbers less than a billion quintillion.
Now suppose a hundred years passes and some clever grad student shows that my claim fails. It first fails at a billion quintillion, and then at every number thereafter.
Obviously no one would care.
Polya's Conjecture
Here's a much cooler example. Recall that prime factorization is the act of writing a number out in terms of its prime factors, which are unique. For example, 15 = 3 times 5, and 3 and 5 are both prime, so that's 15's prime factorization. Moreover, no other pair of primes will work for 15.
As another example, 18 = 2 times 3 times 3, so it's prime factors are 2 and 3. Importantly, though, 3 appears twice, making it so that 18 has three prime factors.
One last example: 360 = 2 times 2 times 2 times 3 times 3 times 5, and therefore 360 has six prime factors.
So, as we see, some numbers have an even number of prime factors (like 15 and 360), and some numbers have an odd number of prime factors (like 18).
Now suppose we created two teams – the even team and the odd team. Let's start at the number 2, and see if it has an odd number of prime factors, or an even number. Whichever case it is, we add a point to that team and then move on to the next number. For the case of 2, the only prime factor is 2, so there's one of them, and one is odd, so one point for the odd team. Current score is 1-0, odd team's winning.
The next number up is 3. 3 has one prime factor – itself – and one is odd, so now the score is 2-0, odd team. Next number up is 4. 4 has two prime factors – 2 and 2 – and since two is even, one point for the even team. Score is 2-1, odd team. Next up is 5, which gives a point to the odd team, so it's 3-1 odd team's lead. 6 gives a point to even, now it's 3-2, still odd team's lead.
You can continue this. As numbers get bigger, you'll find that it's more taxing to compute their prime factorization, but so be it. What you'll find is that the even team never takes the lead.
At least, that's what you'll find if you're doing all this with pen and paper, painstakingly computing the prime factorization of each new number and counting up how many you get. That's exactly what George Polya had done in the early 20th century, and it led him to conjecture that the even team will never take the lead – ever. This became known as Polya's Conjecture, and its Truth (or lack thereof) remained unknown for about 40 years.
However, it turns out that the even team does eventually take the lead. In fact, the first time it takes the lead is at 906,150,257 — a shockingly small number in the grand scheme of things. As "small" as that number appears, it was certainly inaccessible with the technology that Polya had available to him — recall that you'd have to compute the prime factorization for every number up to that number.
Out of curiosity I programmed a simple thing to test this on my (modern) laptop, using a pretty efficient number-of-prime-factors-finding algorithm in python. I stopped the program at about 10,000,000 because I could tell it was going to take about a day to finish and no one has time for that. Therefore, 906,150,257 is still a pretty darn big number, not only from a human perspective but even for a modern laptop.
However, it's still just dust in the wind compared to the enormity of infinity. We might be tempted to think that "the odd team held a lead for a long time, almost making it to a billion (!) before giving up the lead," but we have to fight that temptation. The even team was really only losing for the tiniest blink of an eye.
Let's learn from Polya's mistake (and from his not-mistakes too — he was a tremendous mathematician, this is not meant to throw shade) and not trust numbers. If someone presents to you a fancy result that has been checked up to some "large" number N, please don't place bets. And take some time to really appreciate how, no matter how unfathomably large of an N you can think of, it is still just dust in the wind.