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Topic: ----- ----- ----- conjecture on a diophantine equation
Replies: 1   Last Post: Mar 26, 2013 12:47 AM

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Deep Deb

Posts: 420
Registered: 12/6/04
----- ----- ----- conjecture on a diophantine equation
Posted: Mar 24, 2013 9:56 PM
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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 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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