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Decomposition of an equation
Posted:
Jun 16, 2013 8:40 PM


Consider the equation (1) below.
z^2  y = x^2 (1)
Conditions: z, y, x are coprime integers , y is even and nonsquare
From (1) one gets (2) where
(z + y^1/2)*(z  y^1/2) = x^2 (2).
(2) can then be decomposed as (3) and (4)
z + y^1/2 = u^2 (3) z  y^1/2 = v^2 (4); x^2 = u^2v^2 (5)
(3) and (4) can be consistent if u ^2 = a + b^1/2 (6) and v^2 = a  b^1/2 (7); a^2 > b.
Question: Are the equations of (1) valid? If not why not?
Any helpful comments will be appreciated.



