In the PBS NOVA special Brian Green’s The Elegant Universe, part of the unification of all four forces relied on an equasion equalling 496.

Other than this number being a perfect number, that is, a number that is the sum of its divisors, why is this number critical to unifying gravity with quantum mechanics?

The short answer is, I suppose, that both the SO(32) and E[sub]8[/sub]xE[sub]8[/sub] Lie groups have 496 generators.

The slightly longer answer, while all keeping the technical complications hidden, is that naively there were lots and lots of possible string theories that Green and Schwarz could possibly consider at the time. However, the expectation at the time was that all of these would be inconsistent. In particular, it was already known that string theories suffer from something called anomalies. As I briefly explained in this old thread, these crop up all the time in quantum field theories and, depending on context, can be a really bad thing, revealing that your theory is internally inconsistent. That was the case in string theory.
However, as I’ve said, there was a vast choice of possible particular string theories to consider. One thing you could change to get a different theory was change the Lie group that was involved. There are infinitely many such groups, but there’s a simple standard classification of them. For any particular Lie group, you can think of the elements defining a continuous transformation that depends on d real numbers. The integer d is called the dimension of the group and is the number of generators the group has.
What Green and Schwarz were checking was whether there are anomalies in all string theories of a type called heterotic, the significance of which we’ll skip over. What their calculation essentially showed was that the anomalies in any such theory were proportional to d - 496. Thus string theories involving groups with d = 496 would have no anomalies. And the two Lie groups that had this property were SO(32) and E[sub]8[/sub]xE[sub]8[/sub] (for reasons related to the fact that 496 = 32 x 31/2).

This was big news because, although most possible string theories were still inconsistent, Green and Schwarz could now point to two examples that were anomaly free. That was a huge surprise to most other physicists at the time and singled these particular string theories out as unusual compared to most quantum field theories. Hence the explosion of interest in them when they published this result (the “1st String Theory Revolution”). Other similar examples were then found.

As to why the number 496 was singled out in this way, that’s a lot murkier. The number emerges from calculating anomalies in a rather opaque way: there’s a specific long calculation they had to do and this happens to be the number that comes out at the end.
Furthermore, with the advent of M-theory, the dominant belief amongst the experts has become that there’s nothing that special about the SO(32) and E[sub]8[/sub]xE[sub]8[/sub] string theories. Instead, these are particular manifestations of some deeper M-theory. If that is ever unveiled and understood, then presumably the chain of significance becomes reversed: M-theory will explain why these two examples are special and hence show why anomaly cancellation in heterotic strings requires that d = 496.

Bonzer: If your post above is typical, you are a terrific writer of difficult subjects for a non-specialist audience. If I had any say in the matter, you’d do a cameo (or more) for the Straight Dope staff.

I think this may be unrelated (maybe not), but I remember Richard Feynman always contended that an explanation for the number 137 (the fine structure constant) was necessary for any grand unified theory.