A theorem can be beautiful in many different ways. For me, one of the most striking is when the proof becomes almost trivial provided you look at the problem in the right way. Example; Fermat's "Little Theorem" (high school version). If p is a prime and a is not divisible by p then a^(p-1)-1 is divisible by p. You try various a's and p's and sure enough-. Why should this be? But n years later in grad school you see it's a very special case of the simply proved but elegant Lagrange's Theorem* that the order of a subgroup divides the order of the group (key idea, coset!). I think one's fondness for a theorem depends on how and when one first encountered it, the so called "mathematical experience'. Among my other pets are Pappus's hexagon theorem and the already mentioned fact that a prime is the sum of two squares if and only if it is of the form 4n+1. DG * Does anyone know the reference to Lagrange? I thought groups weren't even invented until much later