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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.
Two examples of a and b with p = ab + 1 | a^2 + b^2
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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 there other examples for a and b for which p | a^2 + b^2 ?
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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
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
> Infinitely many examples. Let a be any positive integer and b = a^3.
> Are there other examples for a and b for which p | a^2 + b^2 ?
30 8 241
But the quotiont is a perfect square also in all these examples.
--
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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
http://en.wikipedia.org/wiki/Vieta_jumping
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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
> > > > Show that a^2 + b^2 divided by ab + 1 is a perfect square.
If p = ab + 1 | a^2 + b^2 and b = na, then n = a^2.
> > Are there other examples for a and b for which p | a^2 + b^2 ?
> Here are some (a, b, p)
> But the quotiont is a perfect square also in all these examples.
What have you tried? I tried working with a and k = b - a > 0.
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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.
> http://en.wikipedia.org/wiki/Vieta_jumping
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