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From: cjh@petsd.UUCP (Chris Henrich)
Newsgroups: net.math
Subject: Re: A number theory problem
Message-ID: <628@petsd.UUCP>
Date: Mon, 26-Aug-85 17:07:10 EDT
Article-I.D.: petsd.628
Posted: Mon Aug 26 17:07:10 1985
Date-Received: Wed, 28-Aug-85 08:44:15 EDT
References: <388@aero.ARPA> <946@oddjob.UUCP>
Reply-To: cjh@petsd.UUCP (PUT YOUR NAME HERE)
Organization: Perkin-Elmer DSG, Tinton Falls, N.J.
Lines: 49
Summary: 

[]
In article <946@oddjob.UUCP> matt@oddjob.UUCP (Matt Crawford) writes:
>...          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?

This kind of question has been extensively studied by number
theorists.  References: many many textbooks on number theory.
That by Hardy (and Littlewood?) is a standard reference.
There is also Cohn, _A_Second_Course_in_Number_Theory_ and
Serre _Arithmetic_ (don't be fooled, this is *advanced*).

Here is an ad-hoc proof of yor observation, based on Gaussian
integers (i.e. complex numbers, whose real and imaginary parts
are integers).

               _
     221 = Z * Z


where    Z = (14 + 5 i)

              _
     5 =  W * W

where
         W =  ( 2 + i ) 

therefore
                        _
     221 * 5^n   =  U * U


where
                       _
         U = Z * W^a * W^b

         a + b = n.

Now the real and imaginary parts of any such U, in either
order, are a solution to your equation.
Regards,
Chris

--
Full-Name:  Christopher J. Henrich
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