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Topic: Formal Proof Language Example - Human-Readable?
Replies: 39   Last Post: Jun 27, 2009 11:09 PM

 Messages: [ Previous | Next ]
 Andrew Tomazos Posts: 137 Registered: 8/1/05
Re: Formal Proof Language Example - Human-Readable?
Posted: Jun 27, 2009 9:29 PM

On Jun 27, 12:52 pm, Jan Burse <janbu...@fastmail.fm> wrote:
> Jan Burse schrieb:
>

> > Now we can continue working with multi sets. Lets
> > assume we have the multi sets for

>
> >    c + a = p1^s1 * ... * pk^sk = q * n * n
> >    c - a = q1^t1 * ... * qj^sj = q * m * m

>
> > Then q is the intersection of the two multi sets,
> > and the rest follows easily:

>
> >    q = intersect(c + a, c - a)
> >    n = sqrt(c + a / q)
> >    m = sqrt(c - a / q)

>
> > Bye
>
> Oops, I was jumping to conclusions. Are n and m
> necessarily coprime?
>
> In your theorem yes. When I started proving, no,
> I didn't require that.
>
> So the n, m are not unqiuely determined in my
> proof. When n and m are not coprime, then
> there is a factor u, such that
>
>      n = u * v
>      m = u * w
>
> Lets see what happens:
>
>      b = n * q * m = v * (q * u * u) * w
>      c + a = q * n * n = (q * u * u) * v * v
>      c - a = q * m * m = (q * u * u) * w * w
>
> Since the intersection will pick the greatest multi set
> that is in both multi sets, it will pick q * u * u.
>
> So my algorithm will deliver coprime n and m. And I
> now believe the intersect function is simply the gcd.
>
> So maybe we just need some theorems about gcd (greatest
> common divisor) and we are done?
>
> Bye
>
> Here is a computed example:
>
>    a = 5
>    b = 12
>    c = 13
>
>    c + a = 18
>    c - a = 8
>
>    q = gcd(c + a, c - a) = 2
>    n = sqrt(c + a / q) = 3
>    m = sqrt(c - a / q) = 2
>
>    b = n * q * m = 2 * 3 * 2 = 12 (yes)
>    c = q * (n*n + m*m) / 2 = 2 * (9 + 4) / 2 = 13 (yes)
>    a = q * (n*n - m*m) / 2 = 2 * (9 - 4) / 2 = 5 (yes)

Yes in this example (c + a / q) = 9 and (c - a / q) = 4. Both perfect
squares. How do you know that they will be perfect squares in all
cases?
-Andrew.

Date Subject Author
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Jan Burse
6/21/09 Jan Burse
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Andrew Tomazos
6/21/09 Jan Burse
6/21/09 Jan Burse
6/24/09 Andrew Tomazos
6/25/09 MeAmI.org
6/25/09 Jan Burse
6/26/09 Andrew Tomazos
6/27/09 Jan Burse
6/27/09 Andrew Tomazos
6/27/09 Jan Burse
6/27/09 Andrew Tomazos
6/27/09 Joshua Cranmer
6/27/09 Andrew Tomazos
6/21/09 Marshall
6/21/09 Spiros Bousbouras
6/24/09 Tim Smith
6/21/09 Charlie-Boo
6/21/09 William Elliot
6/22/09 MeAmI.org
6/22/09 MeAmI.org
6/23/09 Slawomir
6/24/09 David Bernier
6/24/09 MeAmI.org
6/24/09 MeAmI.org
6/24/09 Andrew Tomazos
6/24/09 Andrew Tomazos
6/25/09 Slawomir