"Lucas, Stephen K - lucassk" <lucassk@jmu.edu> wrote:

At a higher level, he has a whole bunch more information of Egyptian fractions at

https://www.ics.uci.edu/~eppstein/numth/egypt/

Interesting. Thanks. It is perhaps unfortunate that there are so many different ways to represent the same number with EFs, many of them quite long. As such, it seems unlikely that advanced aliens will use EFs as their standard way of representing numbers. Similarly with some of the other suggested ways of representing numbers. I was hoping to find a way as succinct and unique as the usual place value notation, but completely different. Is there any reasonable or interesting subset of the reciprocal positive integers for which all EFs of rational numbers still terminate? (It's always possible to skip the first N of them for any N, but not very reasonable or interesting.) The powers of 2 of course give an almost unique representation of every positive number, but they only terminate for dyadic fractions. (Almost unique because each dyadic fraction can end with infinitely many 0s or with infinitely many 1s.) The primes are an especially poor choice. Almost nothing terminates. For instance 18/55 = 1/5 + 1/11 + 1/29 + 1/541 + 1/30829 + .... I did notice that some primes are much likelier than others to show up in a GPEF (greedy prime Egyptian fraction). Highly composite numbers (A002182) are even worse. Not only does almost nothing terminate, but since the sum of their reciprocals converges to about 2.13, nothing larger that can be represented at all.