Source: Simons Foundation website, Feb 2013
In the early 1970s, the mathematician John McKay made a simple observation. He remarked that
196,884 = 1 + 196,883
3 196,884 is the first interesting coefficient of a basic function in that branch of mathematics: the elliptic modular function.
4 196,883 is the smallest dimension of a Euclidean space that has the largest sporadic simple group (the monster group) as a subgroup of its symmetries.
5 McKay gave a convincing interpretation of the 1 in the formula as well.
What is peculiar about this formula is that the left-hand side of the equation, i.e., the number 196,884, is well known to most practitioners of a certain branch of mathematics (complex analysis, and the theory of modular forms),3 while 196,883, which appears on the right, is well known to most practitioners of what was in the 1970s quite a different branch of mathematics (the theory of finite simple groups).4
McKay took this “coincidence” — the closeness of those two numbers5 — as evidence that there had to be a very close relationship between these two disparate branches of pure mathematics, and he was right! Sheer coincidences in math are often not merely sheer; they’re often clues — evidence of something missing, yet to be discovered.
Related Resource: Quora, Aug 2015
Both 196884 and 196883 were integers that came from important objects in mathematics. 196884 was tied to the -function, which was important in analytic number theory (speaking roughly: using fancy calculus to answer questions about primes and other integers). 196883, on the other hand, was tied to the Monster group, which was an important object in algebra, specifically in the classification of all finite simple groups.
Here’s the key point: there was no reason to suspect that there was anything at all in common between the -invariant and the Monster group. They came from completely different fields of study to solve entirely different kinds of problems. And yet… 196884 = 196883 + 1, as John McKay noticed.