According to the Strong Law of Small Numbers: ‘There aren’t enough small numbers to meet the many demands made of them’. Small examples tend to possess many elegant patterns that do not persist once they grow in size. While mathematics is often about pattern seeking, it is equally important (and difficult) to avoid those that are misleading.
Take any positive integer greater or equal to 2. The fundamental theorem of arithmetic says that this number has a unique prime factorisation. For example, 180 = 2 x 2 x 3 x 3 x 5, and all the factors on the right-hand side are primes. The Pólya conjecture states that at least half of the numbers between 1 and our number will have an odd number of factors in its prime factorization, or in short, prime factors. Let’s test that out with a small example: pick the number 7, say.
- 2, 3, 5 and 7 are primes so they each have one prime factor (themselves!), hence an odd number of them.
- 4 = 2 x 2 and 6 = 2 x 3: these have an even number of prime factors.
- 1 is special: it isn’t divisible by any primes, so it has 0 prime factors – an even number of them.
That makes four numbers between 1 and 7 which have an odd number of prime factors, which is more than half of the seven numbers between 1 and 7. The Pólya conjecture holds for the number 7.
Admittedly the computation is rather tedious, but if you try this out with a reasonable set of numbers, you might be fairly convinced that the Pólya conjecture is true in general. After all, our brains are wired to establish patterns from experience and isolated examples. But if it happens so that you put the number 906,150,257 to test (and I hope you didn’t), the Pólya conjecture would fail! The math conjures to throw our intuition from small examples out the window.
The risk of drawing false conclusions from small examples is humorously referred to as the Strong Law of Small Numbers, a phrase coined by the late mathematician Richard K. Guy in an article of the same name [1]. Its namesake is the antithesis of the Strong Law of Large Numbers (which, to the contrary, is a mathematically rigorous statement). In that article, numerous instances of patterns that appear for small numbers are presented, and in each case the reader is invited to suggest whether the pattern holds true in general. I did no better than random guessing.
This isn’t to say that small examples aren’t useful; they are of tremendous value. Oftentimes in mathematics, a pattern will spring up for small cases, leading to a conjecture that the pattern holds true in general. Sometime, perhaps many years later, a subsequent proof confirms its truth. The danger of generalisation lies in the occasional instance where the pattern is only an artefact of the smallness of our test cases.
Who cares if a pattern only breaks down for crazily large cases? Well, some might refer to how the tiniest mathematical mistakes could lead to real-world catastrophe, and indeed history is marked with incidents of that nature. To be fair, though, most problems of this type will find little to no application anytime soon; mathematicians simply study them for their intrinsic quirkiness. Some also take comfort in the idea that a mathematical truth is the closest thing to a fact.
On a closing note, I’d like to draw your attention to the Collatz conjecture, arguably the most notorious unsolved problem facing the challenge of the Strong Law of Small Numbers. The conjecture is incredibly simple: take your favourite positive integer n. If it’s odd, change it to 3n+1; if it’s even, halve it. Repeat again and again. Will we eventually reach the number 1 no matter our starting number?
At the time of writing this, we are nowhere close to solving the problem. What we do have is an insane amount of evidence: computers have checked that for numbers up to around 3 x 1020, we do eventually reach the number 1. But compared to the infinitely many numbers that exist in the mathematical universe, our evidence means nothing at all.
Reference
[1] Guy, Richard K. “The Strong Law of Small Numbers.” The American Mathematical Monthly, vol. 95, no. 8, 1988, pp. 697–712. JSTOR, www.jstor.org/stable/2322249. Accessed 26 Feb. 2021.
Michael Law
The University of Melbourne
