Date: Sun, 27 Oct 1996 16:53:18 -0800 (PST)
From: Milo Gardner <gardnerm@gaia.ecs.csus.edu>
To: David Eppstein <eppstein@ICS.UCI.EDU>
Subject: Unit Fractions, smallest last term
Dear David and Stan:
I offer this information as a question. Kevin comments that
Richard K. Guy's Unsolved Problems in Number Theory may list an
error or a typo (*).
Since I do no own a copy of Guy's book, could the point
to discussed? Better yet, could Guy's email address be
provided so that Guy can respond to the typo or error point?
Kevin's table cites the smallest last term 'Egyptian' solutions to
1 = 1/a + ... + 1/n
Thanks again for the interesting Egyptian fractions discussions.
Milo Gardner
Sacramento, CA
ksb wrote:
By the way, the optimum expansions of 1 into k distinct unit fractions
for k=3,4,... are as listed below:
k denominators of optimum expansion
--- -----------------------------------------------
3 2 3 6
4 2 4 6 12
5 2 4 10 12 15
6 3 4 6 10 12 15
7 3 4 9 10 12 15 18
8 3 5 9 10 12 15 18 20
9 4 5 8 9 10 15 18 20 24
10 5 6 8 9 10 12 15 18 20 24
11 5 6 8 9 10 15 18 20 21 24 28
12 6 7 8 9 10 14 15 18 20 24 28 30
Richard Guy's book on Unsolved Problems in Number Theory states that the
smallest max denominator for k=3 is 6, and for k=4 is 12, both in agreement
with the above table. However, he then says that the smallest max
denominator for k=12 is 120 (*), whereas the above table shows that it is
actually 30. I suppose it's just a typo in Guy's book. (It may even
be less than 30, because I didn't have the patience to check all
possibilities for k=12, but it's certainly no greater than 30.)
From: ksbrown@seanet.com (Kevin Brown)
Date: Sun, 27 Oct 1996 21:27:08 GMT
Newsgroups: sci.math
Subject: Sum of 12 Distinct Unit Fractions = 1
On page 161 of Guy's excellent book "Unsolved Problems in Number
Theory" (2nd Ed) it discusses expressing 1 as the sum of t distinct
unit fractions. Letting m(t) denote the smallest possible maximum
denominator in such a sum, he notes that m(3)=6 and m(4)=12. These
follow from the optimum 3-term and 4-term expressions
1 = 1/2 + 1/3 + 1/6
1 = 1/2 + 1/4 + 1/6 + 1/12
However, the book goes on to say that m(12)=120. Is this just a typo?
If I've interpreted the definition of m(t) correctly it seems to me
m(12) cannot be greater than 30. Here's a table of the optimum
expansions for t=3 to 12:
t denominators of optimum expansion
--- -----------------------------------------------
3 2 3 6
4 2 4 6 12
5 2 4 10 12 15
6 3 4 6 10 12 15
7 3 4 9 10 12 15 18
8 3 5 9 10 12 15 18 20
9 4 5 8 9 10 15 18 20 24
10 5 6 8 9 10 12 15 18 20 24
11 5 6 8 9 10 15 18 20 21 24 28
12 6 7 8 9 10 14 15 18 20 24 28 30
I'm not actually certain the above expansion for t=12 is optimum, but
it proves that the maximum denominator of the optiumum expansion is
certainly no greater than 30. Am I missing something?
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