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Topic: ----- ----- ----- conjecture
Replies: 8   Last Post: Jan 2, 2014 6:12 PM

 Messages: [ Previous | Next ]
 Martin Shobe Posts: 1,469 Registered: 3/11/12
Re: ----- ----- ----- conjecture
Posted: Jan 2, 2014 12:43 PM

On 1/2/2014 10:40 AM, Deep wrote:
> Consider (1) below for the given conditions
>
> z^p - x^p = 2y^2 (1)
>
> Conditions: z, x, y are co prime integers taken two at a time, 2|y, prime p > 3, z > x > y > 0.
>
> Conjecture: (1) can not be satisfied for the given conditions.
>
> Any helpful comment upon the correctness of the Conjecture will be appreciated.
>
> Relevant references about the properties of (1) will be very helpful.
>

Well, z-x | z^p - x^p, so z-x | 2y^2. Therefore z-x | 2 or z-x | y. If
z-x | 2, then x-z | y as 2 | y. Therefore z-x | y.

This means that z-x must divide (z^p - x^p)/(z - x). Dividing (z^p -
x^p)/(z - x) by z-x again leaves a remainder of px^(p-1). Therefore, p =
0 or x = 0. But p > 3 and x > y > 0. So z-x does not divide (z^p -
x^p)/(z - x). Therefore, the equation cannot be satisfied.

Feel free to point out any mistakes.

Martin Shobe

Date Subject Author
1/2/14 Deep Deb
1/2/14 Martin Shobe
1/2/14 quasi
1/2/14 quasi
1/2/14 Martin Shobe
1/2/14 quasi
1/2/14 quasi
1/2/14 Deep Deb
1/2/14 Deep Deb