From: Vladimir Drobot Subject: Re: Diophantine Equation 1/x+1/y+1/z=4/N Date: Fri, 10 Mar 2000 09:30:50 -0800 Newsgroups: sci.math.research,sci.math,rec.puzzles Summary: [missing] There is a book by Mordell, titled Diophantic equations I think, where either the last chapter, or an appendix is devoted to this equation, using the method you suggest. Also, Bill Webb from Western Washington, proved that the density of N's for which it is true is 1. It is enough to consider N to be a prime, and then he used sieve method. This was sometime in the 1970's John R Ramsden wrote: > Does anyone have references for work done on the Egyptian Fraction > problem of proving that for every integer N > 1 there exist positive > integers x, y, z satisfying 1/x + 1/y + 1/z = 4/N? > > I have proved that this is true for every N except possibly when > N = 4.n + 1 and every factor of n + 1 is of the form 12.Z + 1. > > (I have also knocked out a few of the latter exceptions. But the > criteria rapidly become too "sparse" to capture all these pesky > exceptions!) > > Cheers > > --------------------------------------------------------------------------- > John R Ramsden (jr@redmink.demon.co.uk) > --------------------------------------------------------------------------- > The new is in the old concealed, the old is in the new revealed. > St Augustine. > --------------------------------------------------------------------------- ============================================================================== From: John Robertson Subject: Re: Diophantine Equation 1/x+1/y+1/z=4/N Date: Sun, 12 Mar 2000 17:09:58 GMT Newsgroups: sci.math.research,sci.math,rec.puzzles In article <38C9314A.90B06CCC@mathcs.sjsu.edu>, Vladimir Drobot wrote: [as above -- djr] See also Allan Swett's page at http://math.uindy.edu/swett/esc.htm for the largest bound yet established, and Guy's Unsolved Problems in Number Theory, section D11. Sent via Deja.com http://www.deja.com/ Before you buy.