Quadratic Diophantine Theorem
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Newsgroups: alt.math.undergrad, alt.math.recreational, sci.math
From:
JSH <jst... @gmail.com> Date: Fri, 5 Sep 2008 13:00:52 -0700 (PDT)
Local: Sat, Sep 6 2008 4:00 am
Subject: Quadratic Diophantine Theorem
Quadratic Diophantine Theorem: In the ring of integers, given the quadratic expression
c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy
where the c's are constants, for solutions to exist it must be true
((c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3))v^2 + (2(c_2 - 2c_1)*(c_6 -
for some n, where p is any prime coprime to z for a given solution,
v = -(x+y)z^{-1} mod p.
For example with x^2 + y^2 = z^2, I have
c_1 = 1, c_2 = 0, c_3 = 1, c_4 = 1, c_5 = 0, and c_6 = 0
which gives
-4v^2 + 8 = n^2 mod p
for every prime coprime to z, for some n (remember ring is ring of
Making the substitution for v gives
-4(-(x+y)z^{-1})^2 + 8 = n^2 mod p
so
-4(x+y)^2 + 8z^2 = n^2*z^2 mod p
and since x^2 + y^2 = z^2, I can substitute out z, to get
4(x-y)^2 = n^2*z^2 mod p
so the requirement is met, as of course, there are an infinity of
And a square was required here because p can be any prime coprime to a
Notice that the result also applies to the general diophantine
The theorem is proven easily using what I call tautological spaces.
James Harris
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From:
"Dirk Van de moortel" <dirkvandemoor... @nospAm.hotmail.com> Date: Sat, 6 Sep 2008 00:00:44 +0200
Local: Sat, Sep 6 2008 6:00 am
Subject: Re: Quadratic Diophantine Theorem
JSH <jst
... @gmail.com> wrote in message
... @v13g2000pro.googlegroups.com
> Quadratic Diophantine Theorem:
[snip]
> The theorem is proven easily using what I call tautological spaces.
The famous spaces in which whatever pops into Dirk Vdm
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From:
JSH <jst... @gmail.com> Date: Fri, 5 Sep 2008 17:01:44 -0700 (PDT)
Local: Sat, Sep 6 2008 8:01 am
Subject: Re: Quadratic Diophantine Theorem
On Sep 5, 3:00 pm, "Dirk Van de moortel"
<dirkvandemoor
... @nospAm.hotmail.com> wrote:
> JSH <jst
... @gmail.com> wrote in message
> 7ce79450-0709-4634-b6b7-095b7a59e... @v13g2000pro.googlegroups.com
> > Quadratic Diophantine Theorem:
> [snip]
> > The theorem is proven easily using what I call tautological spaces.
> The famous spaces in which whatever pops into
> Dirk Vdm
The only question is, how useful is it?
James Harris
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From:
Rotwang <sg... @hotmail.co.uk> Date: Fri, 5 Sep 2008 21:21:58 -0700 (PDT)
Local: Sat, Sep 6 2008 12:21 pm
Subject: Re: Quadratic Diophantine Theorem
On 5 Sep, 21:00, JSH <jst... @gmail.com> wrote:
> Quadratic Diophantine Theorem:
> In the ring of integers, given the quadratic expression
> c_1*x^2 + c_2*xy + c_3*y^2 = c_4*z^2 + c_5*zx + c_6*zy
> where the c's are constants, for solutions to exist it must be true
> ((c_2 - 2c_1)^2 + 4c_1*(c_2 - c_1 - c_3))v^2 + (2(c_2 - 2c_1)*(c_6 -
> for some n, where p is any prime coprime to z for a given solution,
> v = -(x+y)z^{-1} mod p.
> [...]
> The theorem is proven easily using what I call tautological spaces.
z^{-2} ((c_2 - 2*c_1)x - (c_2 - 2*c_3)y - (c_6 - c_5)z)^2
which is clearly a square. But I don't see what the point of this is,
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From:
Tim Smith <reply_in_gr... @mouse-potato.com> Date: Fri, 05 Sep 2008 22:04:09 -0700
Local: Sat, Sep 6 2008 1:04 pm
Subject: Re: Quadratic Diophantine Theorem
In article <t_hwk.629411$7f3.176... @newsfe23.ams2>, ... @nospAm.hotmail.com> wrote:
> JSH <jst
... @gmail.com> wrote in message
> 7ce79450-0709-4634-b6b7-095b7a59e
... @v13g2000pro.googlegroups.com
> > Quadratic Diophantine Theorem:
> [snip]
> > The theorem is proven easily using what I call tautological spaces.
> The famous spaces in which whatever pops into
--
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From:
Rotwang <sg... @hotmail.co.uk> Date: Fri, 5 Sep 2008 22:12:34 -0700 (PDT)
Local: Sat, Sep 6 2008 1:12 pm
Subject: Re: Quadratic Diophantine Theorem
On 6 Sep, 06:04, Tim Smith <reply_in_gr... @mouse-potato.com> wrote:
> In article <t_hwk.629411$7f3.176
... @newsfe23.ams2>,
> "Dirk Van de moortel" <dirkvandemoor
... @nospAm.hotmail.com> wrote:
> > JSH <jst... @gmail.com> wrote in message ... @v13g2000pro.googlegroups.com
> > [snip]
> > > The theorem is proven easily using what I call tautological spaces.
> > The famous spaces in which whatever pops into
> Do you happen to have a counterexample to his alleged theorem, or some
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From:
"Dirk Van de moortel" <dirkvandemoor... @nospAm.hotmail.com> Date: Sat, 6 Sep 2008 10:37:02 +0200
Local: Sat, Sep 6 2008 4:37 pm
Subject: Re: Quadratic Diophantine Theorem
Tim Smith <reply_in_gr
... @mouse-potato.com> wrote in message
... @news.supernews.com
> In article <t_hwk.629411$7f3.176
... @newsfe23.ams2>,
> "Dirk Van de moortel" <dirkvandemoor
... @nospAm.hotmail.com> wrote:
>> JSH <jst... @gmail.com> wrote in message ... @v13g2000pro.googlegroups.com
>> [snip]
>>> The theorem is proven easily using what I call tautological spaces.
>> The famous spaces in which whatever pops into
> Do you happen to have a counterexample to his alleged theorem, or some
http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/ArmyMath.html Dirk Vdm
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From:
Joshua Cranmer <Pidgeo... @gmail.com> Date: Sat, 06 Sep 2008 09:28:19 -0400
Local: Sat, Sep 6 2008 9:28 pm
Subject: Re: Quadratic Diophantine Theorem
Dirk Van de moortel wrote:
His non sequiturs are half the reason I haven't kill-filed him. The In any case, that posting is quite enlightening, given conversations
Also, JSH being relatively empty from your list of... postings does seem
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From:
Frederick Williams <frederick.willia... @tesco.net> Date: Sat, 06 Sep 2008 15:49:32 +0100
Local: Sat, Sep 6 2008 10:49 pm
Subject: Re: Quadratic Diophantine Theorem
JSH wrote:
> Nope. The theorem is absolutely correct.
> The only question is, how useful is it?
--
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From:
"Dirk Van de moortel" <dirkvandemoor... @nospAm.hotmail.com> Date: Sat, 6 Sep 2008 16:54:08 +0200
Local: Sat, Sep 6 2008 10:54 pm
Subject: Re: Quadratic Diophantine Theorem
Joshua Cranmer <Pidgeo
... @gmail.com> wrote in message
... @aioe.org
> Dirk Van de moortel wrote:
>> I never even *look* at anything between his opening and closing
>> line anymore.
>> Not after this:
>>
http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/ArmyMath.html > His non sequiturs are half the reason I haven't kill-filed him. The
> In any case, that posting is quite enlightening, given conversations
> Also, JSH being relatively empty from your list of... postings does seem
Dirk Vdm
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From:
JSH <jst... @gmail.com> Date: Sat, 6 Sep 2008 08:15:39 -0700 (PDT)
Local: Sat, Sep 6 2008 11:15 pm
Subject: Re: Quadratic Diophantine Theorem
On Sep 6, 7:49 am, Frederick Williams <frederick.willia... @tesco.net>
> JSH wrote:
> > Nope. The theorem is absolutely correct.
> Show us the proof then!
James Harris
