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Perfect Square
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More options Oct 27 2012, 10:25 pm
Newsgroups: sci.math
From: William Elliot <ma...@panix.com>
Date: Sat, 27 Oct 2012 19:25:45 -0700
Local: Sat, Oct 27 2012 10:25 pm
Subject: Perfect Square
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.

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More options Oct 27 2012, 11:27 pm
Newsgroups: sci.math
From: William Elliot <ma...@panix.com>
Date: Sat, 27 Oct 2012 20:27:21 -0700
Local: Sat, Oct 27 2012 11:27 pm
Subject: Re: Perfect Square

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 ?

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More options Oct 28 2012, 5:07 pm
Newsgroups: sci.math
From: Helmut Richter <hh...@web.de>
Date: Sun, 28 Oct 2012 22:07:51 +0100
Local: Sun, Oct 28 2012 5:07 pm
Subject: Re: Perfect Square

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

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More options Oct 29 2012, 4:12 am
Newsgroups: sci.math
From: Robin Chapman <zen165...@zen.co.uk>
Date: Mon, 29 Oct 2012 08:12:42 +0000
Local: Mon, Oct 29 2012 4:12 am
Subject: Re: Perfect Square
On 28/10/2012 02:25, 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.

http://en.wikipedia.org/wiki/Vieta_jumping

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More options Oct 29 2012, 4:24 am
Newsgroups: sci.math
From: William Elliot <ma...@panix.com>
Date: Mon, 29 Oct 2012 01:23:57 -0700
Local: Mon, Oct 29 2012 4:23 am
Subject: Re: Perfect Square

On Sun, 28 Oct 2012, Helmut Richter wrote:
> On Sat, 27 Oct 2012, William Elliot wrote:

> > > > 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.

All variables are positive integers.

If p = ab + 1 | a^2 + b^2 and b = na, then n = a^2.
In addtion, if b = na and n = a^2, then p | a^2 + b^2 = pa^2.

> > 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
...
>  81090   21728   1761923521

Computer generated?

> 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.

What have you tried?  I tried working with a and k = b - a > 0.

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More options Oct 30 2012, 5:59 am
Newsgroups: sci.math
From: William Elliot <ma...@panix.com>
Date: Tue, 30 Oct 2012 02:59:07 -0700
Local: Tues, Oct 30 2012 5:59 am
Subject: Re: Perfect Square

On Mon, 29 Oct 2012, Robin Chapman wrote:
> On 28/10/2012 02:25, 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.

Hurrah.  What a mess.