Cris Moore <moore@santafe.edu> wrote:

The Egyptians were ok with integers, so I think they would have called one thousand one thousand,

I'm less interested in what the actual Egyptians did than in the abstraction loosely based on it. For instance they also allowed 2/3.

as opposed to 1+1/2+1/3+?1/e^1000 :-)

Shouldn't gamma be in there somewhere? Also, e^1000 is not an integer.

But it?s a lovely exercise that the greedy algorithm for Egyptian fractions ? that is, subtract the largest reciprocal ? terminates for any rational number between 0 and 1.

(Note that non-ASCII characters, including "smart" quotes, are turned into question marks on this list, at least on the digest option.) Are you implying that it doesn't always terminate for rational numbers greater than 1? If so, can you give me an example? Thanks. For "random" irrational numbers, the expected distribution of terms in distributed fractions are well understood. What about the expected distribution of terms in greedy Egyptian fractions? Obviously, there can't be a non-zero asymptotic density, as that wouldn't converge. My impression from looking at a few of them is that each term is roughly the square of the previous. That raises the question of whether there's anything special about 1/2 + 1/4 + 1/16 + 1/256 + 1/65536 + ..., and about whether any number has terms that consistently grow more slowly, e.g. each term is on average only the 1.5 power of the previous, forever.