Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site lsuc.UUCP
Path: utzoo!lsuc!msb
From: msb@lsuc.UUCP (Mark Brader)
Newsgroups: net.math
Subject: Re: A number theory problem
Message-ID: <779@lsuc.UUCP>
Date: Tue, 3-Sep-85 13:40:56 EDT
Article-I.D.: lsuc.779
Posted: Tue Sep  3 13:40:56 1985
Date-Received: Tue, 3-Sep-85 13:55:46 EDT
References: <388@aero.ARPA> <770@lsuc.UUCP>
Reply-To: msb@lsuc.UUCP (Mark Brader)
Organization: Law Society of Upper Canada, Toronto
Lines: 32
Summary: 1729 is Carmichael

> The interesting thing here is that 1729 is even more interesting than
> Ramanujan mentioned.  It is not the smallest number to have a certain
> other property, but it IS the THIRD-smallest, and that makes the property
> pretty rare.  Combining this property with the UNRELATED one that
> Ramanujan mentioned makes 1729 very interesting indeed!
> What property am I talking about?

The answer is: 1729 is also a Carmichael number.

Fermat proved that b^p - b is divisible by p for all positive(?) integers b,
if p is prime.  But it's "if", not "if any only if".  A composite number p
having the same property is called a Carmichael number.
According to a program I ran, the ones below 65536 are:

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

My reference on this is the December 1982* Scientific American, the article
on "The Search for Prime Numbers".  (The article said that 561 was the first
of these numbers and then casually mentioned 1729 as another.  I liked that.)

Mark Brader
*Oops, I should have written down the year.  It might be 1983.