From:mathwft@math.canterbury.ac.nz (Bill Taylor)Date:13 Nov 1996 03:10:17 GMTNewsgroups:sci.mathSubject:MORE Egyptian fraction queries.

Egyptian fractions seem to be in, at the moment. I was checking out Guy's UPiNT and saw this conjecture:- All m Exists N n>N ==> Exist x,y,z m/n = 1/x +- 1/y +- 1/z. Note the plus-or-minuses. So we're in Z rather than N. Guy observes that it may be that N > m; and quotes instance N=23 for m=18. So presumably this means 18/23 CANNOT be expressed as the sum/diff of 3 unit fractions. And is (presumably) the "first" counterexample. Query: Is there any reason for 18/23. I guess not; it just seems to be a random-looking phenomenon. No doubt there are many others in this much-conjectured but imperfectly known field; I bet the law of small numbers has great sway in Egyptian fractions! Query: What are the next few that cannot be so expressed. Can some of those newsgroupies who've announced the possession of fast software for unit fractions, post a list of examples of other failures; up to some convenient denominator size. That is, fractions m/n with m<n, that cannot be written 1/x +- 1/y +- 1/z. Thanks! ------------------------------------------------------------------------------- Bill Taylor W.Taylor@math.canterbury.ac.nz ------------------------------------------------------------------------------- This stupid computer never does what I want it to, only what I tell it to! -------------------------------------------------------------------------------