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From: pumphrey@ttidcb.UUCP (Larry Pumphrey)
Newsgroups: net.math
Subject: Re: A number theory problem
Message-ID: <447@ttidcb.UUCP>
Date: Wed, 28-Aug-85 12:33:05 EDT
Article-I.D.: ttidcb.447
Posted: Wed Aug 28 12:33:05 1985
Date-Received: Fri, 30-Aug-85 00:38:00 EDT
References: <388@aero.ARPA>, <946@oddjob.UUCP>
Organization: TTI, Santa Monica, CA.
Lines: 40


> Misspending part of a summer on this sort of question led me to
> observe that the number 221*5^n seems to be expressible as a sum
> of two squares 2n+2 different ways for n through at least 10 or
> so.  Is it true for all n?  If so, why?

Yes, it is true for _all_ n.  In fact the more general problem can be
phrased as follows:

			    n
			_________
	    |\    |       |   |     |/
	    | \   |       |   |   _ |\i
   Let      |  \  | =     |   |  |_)
	    |   \ |       |   |  |  i
	    |    \|       |   |
			   i=1


   where each p  is a prime of the form 4x+1 and further
	       i
   let f(N) be the number of _different_ divisors of N (1 is
   considered a divisor) less than or equal to the square root of N.

   Then N can be represented as the sum of 2 squares in exactly f(N) ways.

   -----------------------------------------------------------------------

   First proof wins a box of cheerios.  My proof is over 200 lines long so
   I'm not posting at this time --- margin is too small to contain it  :-)
   Hint: It can be solved by elementary means, that is to say algebraically
	 rather than analytically!

   p.s. Don't worry about misspending a summer on this problem, I've
	wasted my whole life (48 and still counting) trying to prove

			x^n + y^n = z^n

	has no integral solutions for n>2   :-(
						   Enjoy!