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From: gmf@uvacs.UUCP
Newsgroups: net.math
Subject: Interesting Numbers
Message-ID: <1274@uvacs.UUCP>
Date: Thu, 3-May-84 10:50:02 EDT
Article-I.D.: uvacs.1274
Posted: Thu May  3 10:50:02 1984
Date-Received: Sat, 5-May-84 01:04:20 EDT
Lines: 27


>  ... it can be proved that all numbers are interesting.  Can
>  anyone give a short proof?  Identify the flaw in such a proof,
>  if any??

If the set of non-interesting positive integers is non-empty, it has
a least element which, as such, is interesting.  Hence the set of
non-interesting positive integers is empty, so all positive integers
are interesting.

Maybe a number can be both interesting and non-interesting.  Are
non-interesting and uninteresting the same?

>  Can anyone prove the following are interesting numbers:
>  (a)  17     (b) 100     (c) 1729     (d) 127

(a)  17 is the smallest arbitrary integer

(b)  100 is the smallest number with 3 digits

(c)  1729 is Ramanujan's number (in Hardy's anecdote):  the smallest
     positive integer that can be represented as a sum of cubes in
     two different ways

(d)  127 is the smallest counterexample to the proposition that all
     odd numbers >= 3 can be represented as a sum of a prime and a
     power of 2 (this was just established in net.math)