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]