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.