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

I found an APL program to compute unit fraction

that found the following answer shown below

Dan Hoey wrote -------

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

APL ans (2346193812419340 33397845 4484 45 7 3 2)

Using PC FP good to about 15 places So largest number is only good

to 2 places. The program used comes from

a 1991 System Journal - the program is short and produces the result

with no visable delay. On the other hand it produces one result not other

ones 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- ÃÂÃÂ·Z <- ceiling ÃÂÃÂ·X

[4] * EX 1E-16 FUN 335ÃÂÃÂ·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 - ÃÂÃÂ·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 335ÃÂÃÂ·336 yeilds 2 3 7 48

which stands for 1/2 1/3 1/7 1/48

monadic divide ÃÂÃÂ· takes one over the each element in a vector

In another post I was wrong about the period of 732/733

Helmut Richter has a web site which shows that it is 61

All rational fractions are finite repeating decimals

Dan 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

>