Egyptian fraction representations of real numbers I've been looking into Egyptian fractions now, but information on certain topics seems scarce. Can you answer any of these questions that intrigue me:
1) What is known about the Egyptian fraction representation (by the greedy algorithm) of irrational numbers? Are Egyptian fractions known to be interesting in any similar sense as the continued fractions? I have not found any references to these expansions; OEIS lists a few terms for some numbers though.
2) What would be the "average" growth of the n:th term in the sequence? The slowest possible would be 2, 3, 7 etc. but for a random real, what is the expected size of the n:th term?
 A: There's a little bit of info here:
 H. E. Salzer, The Approximation of Numbers as Sums of Reciprocals,
American Mathematical Monthly, Vol. 54, No. 3 (Mar., 1947), pp. 135-142.  There are very few "naturally-occcurring" real numbers for which their expansion is known.  One example where it is known is for (3-sqrt(5))/2, where the denominators are given by the (2^n)th Fibonacci number.
A: After trying to parse the following painstakingly-simple algorithm into plain English, I gave up all hope:
$$N = \underbrace{\underbrace{\underbrace{\underbrace{\underbrace{\underbrace{[N]}_{A_0}+\frac1{\left[N-A_0\right]}}_{A_1}+\frac1{\left[N-A_1\right]}}_{A_2}+\frac1{\left[N-A_2\right]}}_{A_3}+\frac1{\left[N-A_3\right]}}_{A_4}+\frac1{\left[N-A_4\right]}}_{A_5}+\ldots$$
If the terms are not allowed to oscillate ($\pm$) , then, with the exception of [N] , [...] should stand for upper-rounding. However, if fast convergence is what's desired, [...] should stand for rounding. In any case, continued fractions make for better convergents than the Egyptian ones under the same conditions/restrictions/limitations.
The famous Hungarian—Jewish mathematician Erdos devoted a significant part of his life to the study of Egyptian fractions. See: Greedy algorithm for Egyptian fractions , Odd greedy expansion , Erdos-Graham conjecture , and Erdos-Straus conjecture.
