I don't know -- very interesting question. But I wanted to chime in that 24 is my choice for most interesting integer, too. Partly because, among all the multiplicative groups of rings G_n := (Z/nZ)*, 24 is the largest n for which k | n implies that all elements of G_k are squares. Also because of its appearance in Dedekind eta. (Are the two things directly related?) --Dan << There are at least three solutions to the Diophantine equation: m² = n! + 1 namely (m,n) = (5,4), (11,5), and (71,7). As far as I know, no further pairs have been discovered. Is there a proof that these three solutions are the only ones in existence, or could there be others? 71 seems to exhibit other interesting properties, being the largest prime factor of the order of the Monster Group, and the largest supersingular prime. But 24 definitely wins as the most interesting integer...
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