Hello everybody, I recall a remark from Hardy and Wright (in Introd. to the theory of numbers) is about this : the only exception is (as we know) the number exp(1). There are no other results known in that direction with n!. That is , the fractional part of exp(1)*n! slowly decrases to 0 as a consequence the expansion of {exp(1)} in factorial base is [1,1,1,1,...] and the expansion of X (any real number) into factorial base never gave any interesting example apart from numbers like exp(1). Another example is {exp(1/2)} into factorial base does not provide any clue either : [1, 0, 1, 0, 3, 2, 5, 0, 4, 3, 9, 8, 2, 8, 0, 10,...] So, the only workable examples are either pathologic or we know nothing about it. In my opinion, maybe the problem is tractable if we can charactetize the sequences obtained from the expansion of X into factorial base. There is a short article on that subject in the Collected Papers of W. Sierpinski which is interesting, see vol III, the version I have is in french. These are difficult matters to settle, for example the problem of {(3/2)^n} as not yet been resolved. The only thing we have are bounds, there has been some progress but it is still a unsolved problem. Unless someone know any example ? Simon Plouffe