```Date: Mar 24, 2013 9:56 PM
Author: Deep Deb
Subject: -----   -----   ----- conjecture on a diophantine equation

Consider the following equation under the given conditions.x^2k + y^2k = z^2k             (1)Conditions: x, z are coprime odd integers, prime k > 3Conjecture: y^k = U^(1/2) where U is a non-square integer.Justification of the Conjecture.(1) is the Fermat's equation for even exponent. Therefore,in (1) if x and z are integers y is not an integer.Let x = uv  where u = a + b^(1/2)        (2)       v = a - b^(1/2)             (3)a, b are positive integers and b is non-square.(1) can then be decomposed into (4) and (5)z^k + y^k = u^2k          (4)         z^k - y^k  = v^2k                   (5)From (4) one gets (6)  where y^k =[ a + b^(1/2) ]^2k -z^k         (6)It can now be argued that for certain values of a, b, z it is possible to obtain (7) from (6)y^k = U^1/2      (7)   where U is a non-square integer.Similar result is obtained by considering (5).This justifies the conjecture. Question: Is (7) correctly derived? If not where is the error? Is the conjecture valid?Any helpful comment upon the validity of the conjecture will he gratefully appreciated.
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