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Topic: -----Decomposition of an equation
Replies: 2   Last Post: Jun 17, 2013 7:41 PM

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Deep Deb

Posts: 420
Registered: 12/6/04
-----Decomposition of an equation
Posted: Jun 16, 2013 8:40 PM
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Consider the equation (1) below.

z^2 - y = x^2 (1)

Conditions: z, y, x are coprime integers , y is even and non-square

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.

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