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Topic: Egyptian fraction identities
Replies: 3   Last Post: May 26, 2013 2:05 PM

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James Waldby

Posts: 374
Registered: 1/27/11
Re: Egyptian fraction identities
Posted: May 26, 2013 11:26 AM
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On Sun, 26 May 2013 10:23:59 -0400, David Bernier wrote:

> On 05/26/2013 02:10 AM, David Bernier wrote:
>> John Baez on his Wordpress blog had an interesting fact about
>> 42 today:

[big snip]
> If 1/2 = 1/p + 1/q + 1/r + 1/s + 1/t
>
> with p, q, r, s, t positive integers such that
> p <= q <= r <= s <= t,
> one possibility is given by:
> 1/2 = 1/3 + 1/8 + 1/31 + 1/124 + 1/744 .
> [ p=3 , q=8, r=31, s=124, t=744. ]
>
> Is it possible to have t > 744 ?


Possibilities include sums like
1/3+1/8+1/26+1/313+1/97656
1/3+1/7+1/44+1/925+1/854700
1/3+1/7+1/43+1/1807+1/3263442

The last of those examples probably is extreme, since
it is the result of a greedy approach; note that all of
1/3+1/6
1/3+1/7+1/42
1/3+1/7+1/43+1/1806
add up to 1/2.

> This question is motivated by John Baez'Puzzle 1.
> at his Azimuth Wordpress blog here:
> < http://johncarlosbaez.wordpress.com/ >
>
> --> 42
>
> Puzzle 1.
> ``Consider solutions of 1/p + 1/q + 1/r = 1/2
> with positive integers p <= q <= r , and
> show that the largest possible value of r is 42. "


--
jiw



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