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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
Re: -----Decomposition of an equation
Posted: Jun 17, 2013 7:41 PM

On Sunday, June 16, 2013 8:40:13 PM UTC-4, Deep wrote:
> Consider the equation (1) below.
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> z^2 - y = x^2 (1)
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> Conditions: z, y, x are coprime integers , y is even and non-square
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> From (1) one gets (2) where

Question: ***** *****
Can y^2 be an integer > 1?

***** *****
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> (z + y^1/2)*(z - y^1/2) = x^2 (2).
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> (2) can then be decomposed as (3) and (4)
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> z + y^1/2 = u^2 (3) z - y^1/2 = v^2 (4); x^2 = u^2v^2 (5)
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> (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.
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> Question: Are the equations of (1) valid? If not why not?
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