From: kfl@KeithLynch.net   
      
   Gary McGath  wrote:   
   > The question was whether such pairs have a name.  You're saying that   
   > 8 and 9 are the only pair as Joy describes it, so it's likely that   
   > kind of pair doesn't have a name.   
      
   Apparently so.  I checked "Catalan pairs," but that turns out to   
   be the name of a different concept named for the same 19th century   
   mathematician.   
      
   He wasn't actually the first to make the conjecture, just the first to   
   widely publicize it.  It's also called Mihailescu's theorem.  The old   
   name is mostly still used since the proof is so new, dating to 2002.   
      
   Ironically, it was proven just for squares and cubes by Gersonides in   
   1343.  So someone partially solved it half a millennium before Catalan   
   even conjectured it.   
      
   But it was far from the oldest math problem.  Whether odd perfect   
   numbers exist has been speculated about for at least 2400 years.  (A   
   perfect number is a number equal to the sum of its proper divisors.)   
   52 such numbers are known, most of them discovered by exhaustive   
   computer searches in our lifetime, but all of them are even.  Odd   
   perfect numbers have neither been found nor proven impossible -- which   
   is odd indeed.   
      
   It's good to see discussion here on such an interesting topic.   
   Indeed, there's a simple proof that every positive integer is   
   interesting:  If any integers were not interesting, there must   
   be a first one, and that would make that first one interesting,   
   which would be a contradiction.   
      
   This proof can't be generalized to real numbers.  Most real numbers   
   are not only not interesting, but are actually impossible to compute,   
   specify, or even think about.  They have a rather ghostly existence.   
   It's been controversially suggested that only *computable* numbers   
   actually exist.  But that's obviously false.  The Chaitin constants   
   are not computable but are well-defined and have definite values.   
   --   
   Keith F. Lynch - http://keithlynch.net/   
   Please see http://keithlynch.net/email.html before emailing me.   
      
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