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In What's so special about the divisors of $24$? (Math. Mag., 2012) it is noted that the exponent of the group of units modulo $n$, that is the highest order of an element of $U(n):=(\Bbb Z/n\Bbb Z)^\times$, is precisely $2$ if and only if the integer $n$ divides $24$. An elementary argument is given (see also answers here for a one-line proof), as well as some analytic machinery, but I recall (perhaps not quite accurately) that the number $24$ shows up a lot in high-powered math related to number theory, like lattices, moonshine, modular forms, string theory etc: suspicious.

If I were a believer in the magical and skeptical of coincidences, I might want to know if there is a high-powered explanation of this fact from the cited theoretical areas (not including asymptotic or statistical heuristics from analytic number theory). Or is it merely a collision of small numbers?

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    $\begingroup$ I think the explanation might go in the other direction: Gannon in Moonshine beyond the Monster speculates that this property of $24$ is responsible for its appearance in high-powered math. $\endgroup$
    – Qiaochu Yuan
    Oct 19, 2012 at 18:06
  • $\begingroup$ Wikipedia echoes this, stating "This fact plays a role in monstrous moonshine." $\endgroup$
    – anon
    Oct 19, 2012 at 18:09
  • $\begingroup$ $\exp U(n) = \lambda(n)$, where $\lambda$ is the Carmichael function. From that follows an easy explanation. $\endgroup$
    – lhf
    Oct 19, 2012 at 18:24

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I believe I've tracked down what Qiaochu was referencing in Gannon's Moonshine beyond the Monster, pg168-169 §2.5 (alas, five pages too late). I can't claim to understand any of it.

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