At 12:47 PM 11/27/02 -0700, you wrote:
Calculate the natural log of (95536/e)^{95536}.
I'm assuming Richard wouldn't have asked this question if there weren't something profound going on ... and the _real_ puzzle is to figure out what it is. The function f(x) = x (ln x - 1) is the integral of ln x. Its values at integer arguments are spaced by ln x, therefore. And so for one of these values to be an integer plus or minus a small p is a surprise on the order of p/2(ln x); we would expect it to happen every 2(ln x)/p integers in the vicinity of x. Near 10^6, we would expect f(x) to be within .001 of an integer about once every 6000 integers. So, this one-in-6000 chance happens for 10^6. I'm impressed enough that I'm sure there's a deeper principle going on. -A