Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10 beta 3/9/83; site nbs-amrf.UUCP Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!decvax!ittvax!dcdwest!sdcsvax!bmcg!cepu!trwrba!trwrb!sdcrdcf!hplabs!hao!seismo!rlgvax!cvl!umcp-cs!nbs-amrf!hopp From: hopp@nbs-amrf.UUCP Newsgroups: net.math Subject: Re: INTERESTING NUMBERS Message-ID: <273@nbs-amrf.UUCP> Date: Thu, 3-May-84 18:19:34 EDT Article-I.D.: nbs-amrf.273 Posted: Thu May 3 18:19:34 1984 Date-Received: Tue, 8-May-84 00:46:17 EDT Organization: National Bureau of Standards Lines: 58 Reference: <14100005@smu.UUCP> > Let us consider "INTERESTING" numbers: > (let us stick to only positive integers) > . . . > > Can anyone prove that the following are interesting numbers: > > (a) 17 (b) 100 (c) 1729 (d) 127 (a) 17 is interesting For all sorts of reasons. For one thing, it is equal to 3**2+2**3. The 17th of the Jewish month of Tammuz is a fast day to mourn the destruc- tion of the second temple in Jerusalem by the Romans. Durer's magic square: 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16 adds up to twice 17 in every direction, and is composed of all the numbers less than 17. Also, it is the average number of diapers my son wore per day during the first month of his life. (b) 100 is interesting Because it is the shortest (nontrivial) representation of a perfect square in every number base. (c) 1729 is interesting Because G. H. Hardy rode in taxi cab No. 1729 on his way to see Srinivasa Ramanujan at Putney. (Also, because it is the smallest number that can be expressed as the sum of two cubes in two different ways: 10**3+9**3 and 1**3+12**3). (d) 127 is interesting It must be interesting -- see below. (Computer types, of course, recognize the infamous 2**7-1.) > In general, using proof by contradiction it can be proved >that all numbers are interesting. Can anyone give a short proof? Assume that not all numbers are interesting. Then the set of non- interesting numbers is not empty. Since it is not empty, it has a smallest element x. One of the interesting things about x is that it is the smallest element of the set of non-interesting numbers. That makes it interesting. If x is interesting, it cannot be a member of the set of non-interesting numbers -- a contradiction. The only assumption made in arriving at this contradiction was that not all numbers are interesting; thus that assumption must be false. > Identify the flaw in such a proof, if any?? There is no flaw. I never make a mistake. (I once thought I made a mistake, but I was wrong.) -- Ted Hopp UUCP: {seismo,allegra}!umcp-cs!nbs-amrf!hopp National Bureau of Standards ARPA: hopp.nbs-amrf.umcp-cs@udel-relay Metrology A127 BELL: (301)921-2461 Washington, DC 20234