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