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   conjecture on a diophantine equation
Posted:
Mar 24, 2013 9:56 PM


Consider the following equation under the given conditions.
x^2k + y^2k = z^2k (1)
Conditions: x, z are coprime odd integers, prime k > 3
Conjecture: y^k = U^(1/2) where U is a nonsquare 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 nonsquare.
(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 nonsquare 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.



