From: HJSmith@ix.netcom.com (Harry J. Smith)
Newsgroups: sci.math
Subject: 2^239 - 1, is known composite, but . . .
Date: Sun, 03 Nov 1996 22:07:20 GMT
I found the following at URL
ftp://ftp.netcom.com/pub/hb/hbaker/hakmem/hakmem.html
It is about the number 239, but some of it (one thing at least) is not true. It
says that the 239th Mersenne number, 2^239 - 1, is known composite, but no
factors are known. BUT 2^239 - 1 is divisible by 479. What is the explanation of
this error? Is the other information correct?
-----
Massachusetts Institute of Technology
A. I. Laboratory
Artificial Intelligence Memo No. 239
February 29, 1972
HAKMEM
contents index
M. Beeler [beeler@bbn.com]
R. W. Gosper [rwg@newton.macsyma.com]
R. Schroeppel [rcs@cs.arizona.edu]
[Retyped and formatted in 'html' ('Web browser format) by Henry
Baker, April, 1995. The goal of this 'html' document is to make
HAKMEM available to the widest possible audience -- including those
without bitmapped graphics browsers. Therefore, equations have been
formatted to be readable even on ASCII browsers such as 'lynx'. Click
here to get original AI Memo 239 in 400 dots/inch, 1 bit/pixel, Group
4 facsimile TIFF format (a single 5 megabyte gzip compressed tar
file, AIM-239.tiff.tar.gz).]
Work reported herein was conducted at the Artificial Intelligence
Laboratory, a Massachusetts Institute of Technology research program
supported in part by the Advanced Research Projects Agency of the
Department of Defense and monitored by the Office of Naval Research
under Contract Number N00014-70-A-0362-0002.
Reproduction of this document, in whole or in part, is permitted for
any purpose of the United States Government.
ITEM 63 (Schroeppel, etc.):
The joys of 239 are as follows:
* pi = 16 arctan (1/5) - 4 arctan(1/239),
* which is related to the fact that 2 * 13^4 - 1 = 239^2,
* which is why 239/169 is an approximant (the 7th) of sqrt(2).
* arctan(1/239) = arctan(1/70) - arctan(1/99) = arctan(1/408) + arctan(1/577)
* 239 needs 4 squares (the maximum) to express it.
* 239 needs 9 cubes (the maximum, shared only with 23) to express it.
* 239 needs 19 fourth powers (the maximum) to express it.
* (Although 239 doesn't need the maximum number of fifth powers.)
* 1/239 = .00418410041841..., which is related to the fact that
* 1,111,111 = 239 * 4,649.
* The 239th Mersenne number, 2^239 - 1, is known composite, but no factors
are known.
* 239 = 11101111 base 2.
* 239 = 22212 base 3.
* 239 = 3233 base 4.
* There are 239 primes < 1500.
* K239 is Mozart's only work for 2 orchestras.
* Guess what memo this is.
* And 239 is prime, of course.
-----
-Harry
--
| Harry J. Smith, 19628 Via Monte Dr., Saratoga, CA 95070-4522, USA
| Home Phone: 1 408 741-0406, Work Phone: 1 408 235-5088 (Voice Mail)
| E-mail: HJSmith@ix.netcom.com, Fax: 1 408 235-2019
| Web site: http://www.netcom.com/~hjsmith
--
==============================================================================
[I don't know why that statement had been made. 2^239-1=
883423532389192164791648750371459257913741948437809479060803100646309887=
product of these prime factors:
479
1913
5737
176383
134000609
7110008717824458123105014279253754096863768062879
-- djr]