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Re: Egyptian fraction identities
Posted:
May 26, 2013 11:26 AM


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



