```Date: Mar 21, 2000 12:48 PM
Author: Joe Ascoly
Subject: Re: unitary (Egyptian) fractions

I found an APL program to compute unit fractionthat found the following answer  shown belowDan Hoey wrote -------By the way, I found that 732/733 has 2771 different seven-termrepresentations.  The largest denominator appears in therepresentation (2305193137933140 33397845 4484 45 7 3 2).  TheAPL ans           (2346193812419340 33397845 4484 45 7 3 2)Using PC FP good to about 15 places So largest number is only goodto 2 places.  The program used comes froma 1991 System Journal  - the program is short and produces the resultwith no visable delay. On the other hand it produces one result not otherones that also are solutions.      Array language APL program FUN      [0]  Z <- T FUN X      [1]  * SYVESTER ALG      [2]  execute (X<=T)/  '-> 0,0 reshape Z <-iota 0'      [3]  Z <- Z,T FUN X- ÃÂÃÂ·Z <- ceiling ÃÂÃÂ·X      [4]  * EX 1E-16 FUN 335ÃÂÃÂ·336 * YIELDS 2 3 7 48 Comments line 0  function header line 1  a comment this is sylvester algorithm line 2  set up if X is less than or equal the tolerance         set truth to 0 or 1         if false go to 0 which is function exit         if true  initalize Z to a null number         since Z will be concatnated it has to have something         in it to start - namely null line 3  execute right to left         Z gets ceiling of recipical of X         X -  ÃÂÃÂ·Z         call fun recursively and Z gets Z, FUN result         comma is concatenate         Left argument to FUN is the compare tolerance         set to 1E-16 line 4  a comment example call         1 E-16 FUN 335ÃÂÃÂ·336  yeilds 2 3 7 48         which stands for 1/2 1/3 1/7 1/48monadic divide ÃÂÃÂ· takes one over the each element in a vectorIn another post I was wrong about the period of 732/733Helmut Richter has a web site which shows that it is 61All rational fractions are finite repeating decimalsDan Hoey wrote in message <20000320210410.11752.00002552@ng-ci1.aol.com>...>It took about 13 hours on a 296 Mhz Ultra.  If a mips is a>Mhz, that's almost half a mips-year.  I don't how much of the>remaining speed difference is due to the programming language/system>and how much is due to the algorithm.>>By the way, I found that 732/733 has 2771 different seven-term>representations.  The largest denominator appears in the>representation (2305193137933140 33397845 4484 45 7 3 2).  The>smallest maximum denominator appears in (26388 20524 7330 45 7 3 2).>>Dan Hoey <haoyuep@aol.com>               Posted and e-mailed>
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