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From: Helmut Richter <hh...@web.de>
Newsgroups: sci.math
Subject: Re: Perfect Square
Date: Sun, 28 Oct 2012 22:07:51 +0100
Organization: Technische Universitaet Muenchen, Germany
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On Sat, 27 Oct 2012, William Elliot wrote:

> On Sat, 27 Oct 2012, William Elliot wrote:
> 
> > Here's an unsolved problem from Ask An Algebraist.
> > 
> > > Let a and b be two positive integers such that ab+1 divides a^2 + b^2.
> > 
> > > Show that a^2 + b^2 divided by ab + 1 is a perfect square.
> > 
> > Two examples of a and b with p = ab + 1 | a^2 + b^2
> > are (1,1) and (2,8), for which (a^2 + b^2)/p = 1 and 4.
> 
> Infinitely many examples.  Let a be any positive integer and b = a^3.
> Then p = a^4 + 1 divides a^2 + b^2 = a^2 + a^6 = a^2 (a^4 + 1).
> 
> Are there other examples for a and b for which p | a^2 + b^2 ?

Here are some (a, b, p)

    30       8          241
   112      30         3361
   240      27         6481
   418     112        46817
  1020      64        65281
  1560     418       652081
  2133     240       511921
  3120     125       390001
  5822    1560      9082321
  7770     216      1678321
 16256    1020     16581121
 16800     343      5762401
 18957    2133     40435282
 21728    5822    126500417
 32760     512     16773121
 59040     729     43040161
 77875    3120    242970001
 81090   21728   1761923521

But the quotiont is a perfect square also in all these examples.
I have some ideas how one could try to prove it but none of them makes for 
a complete proof.

-- 
Helmut Richter