From: Deinst@world.std.com (David M Einstein) Subject: Re: Egyptian Fractions Date: Sun, 19 Dec 1999 21:57:40 GMT Newsgroups: sci.math Keywords: What numbers are sums of two Egyptian fractions? Obviously, 5/2 cannot be the sum of two egyptian fractions. Not quite so obviously 7/58 is not the sum of two egyptian fractions. In general the sum of two egyptian fractions is (1/g)((x+y)/xy) with x and y relatively prime. To see if p/q is represented as the sum of two unit fractions run through the pairs of relatively prime factors of q and look for a pair that has xy|q and (x+y)|p. For example if q is 58, the factors of q are 1,2,29, and 58, choices for {x,y} are {1,2}, {1,29}, {1,58}, or {2,29}, so (x+y) is one of 3,30,59,or 31, then if p/q is the sum of two unit fractions then p is one of 1,2,3,5,6,10,15,30,31,or 59. For more information on egyptian fractions see http://www.ics.uci.edu/~eppstein/numth/egypt/ Deinst Matthew Burgess (mattb@nospam.umit.maine.edu) wrote: : Hi all, : I have a question regarding egyptian fractions and what we know about : them. I am familiar with the conjecture that for all numbers n > 1, : there are positive integers x, y, and z such that: : 1/x + 1/y + 1/z = 4 / n : Many of these numbers n have solutions in a simpler equation: : 1/x + 1/y = 3 / n : where obviously z = n. It is also clear that this "simpler" equation : is not true for all n > 1, since n = 7 has no positive integer : solutions. : Here's my question: : What kind of numbers can be represented by two egyptian fractions? : Can all fractions in the form (odd number) / (even number) be : represented, and if so, why? It is clear from above that fractions of : the form (odd) / (odd) cannot always be represented (3/7), and it is : also obvious that fractions of the form (even) / (even) cannot always : be represented (6/14). But is there something to be said for the sum : of two positive egyptian fractions? : Thanks in advance, : Matt ============================================================================== From: David M Einstein Subject: Re: Egyptian Fractions Date: Mon, 20 Dec 1999 12:49:40 GMT Newsgroups: sci.math Matthew Burgess wrote: > Most other numbers can be accounted for. All multiples of 3 and of > numbers of the form 3k - 1 all have solutions to: > > 1/x + 1/y = 3/n. > > However, at least one number of the form 6a + 1 does not have a > solution, namely 7. That's what got my interest and wondered about See Dave Eppsteins post. It has a pointer to a proof that 1/x + 1/y = 3/n is solvable iff n has a factor =3 mod 2. In particular primes of the form 6n+1 have no factors =3 mod 2, so are not solvable. > the relationships between (x+y) and (xy). I am interested to see your > reply to my other post regarding numbers a of the form a = 0 (mod 4), > which are numbers n of the form 24m + 1. Another person (I regret not > remembering their name) was dealing with numbers of this type and > egyptian fractions last summer, but I can't remember the thread or his > results. > > I'm a CompSci major in college, trying for my math minor, and I have > discovered a love for amateur number theory. If you do get a solution > for my other post, could you elaborate slightly to help a budding > mathematician out? I have followed most things, but I am still trying > to understand how to analyze the congruences for Q that you listed. > > Thanks again, > Matt