Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site dciem.UUCP
Path: utzoo!dciem!ntt
From: ntt@dciem.UUCP (Mark Brader)
Newsgroups: net.math
Subject: Further notes on the number 1729
Message-ID: <914@dciem.UUCP>
Date: Wed, 9-May-84 12:34:28 EDT
Article-I.D.: dciem.914
Posted: Wed May  9 12:34:28 1984
Date-Received: Wed, 9-May-84 14:24:41 EDT
References: <14100005@smu.UUCP>, <1274@uvacs.UUCP>
Organization: NTT Systems Inc., Toronto, Canada
Lines: 35

I like George Fisher's calling 1729 "Ramanujan's Number"!
Incidentally, some fun is had with the story behind this in Hofstadter's
"Godel, Escher, Bach: An Eternal Golden Braid".  I'll say no more here.

But the really interesting thing about 1729 is that in addition to being
expressible as the sum of two cubes in two different ways, it has ANOTHER
very rare property.  It is a Carmichael number.

Let F(m,n) = (m^n - m) % n, where ^ is exponentiation and % is modulus.
Then if n is prime, F(m,n)=0 for all m.  [Fermat.]  Well, it turns out that
the converse is also true for MOST numbers m and n.  That is, if F(m,n)=0,
n is USUALLY prime.  If it isn't, it is called a pseudoprime to the base m.
If n is composite but F(m,n)=0 for all m, that is, if n is a pseudoprime to
all bases, then n is called a Carmichael number, after R. D. Carmichael.

The above is from an article in Scientific American by Carl Pomerance,
which was in the December 1982 issue.  The article casually mentions 1729
as an example of a Carmichael number without saying why that example was
picked.  Both examples of Carmichael numbers in the article had exactly 3
prime factors.  I wrote a program to find all the Carmichael numbers up
to 2^16, and factorize them, to see if the pattern continued.  Results:

	  561 = 3x11x17
	 1105 = 5x13x17
	 1729 = 7x13x19
	 2465 = 5x17x29
	 2821 = 7x13x31
	 6601 = 7x23x41
	 8911 = 7x19x67
	10585 = 5x29x73
	15841 = 7x31x73
	29341 = 13x37x61
	41041 = 7x11x13x41

Mark Brader