Re: Simultaneous Equations

In article <1177197075.375814.206830@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
Deep <deepkdeb@xxxxxxxxx> writes:
Consider the following two simulteneous equations:

x^6 -21x^4y^2 + 35x^2y^4 - 7y^6 = R^1/2 (1)

y^6 -21y^4x^2 + 35y^2x^4 - 7x^6 = S^1/2 (2)

Given: R and S are nonsquare integers such that (R, S) = 1, R is even.

Assertion: x^2 + y^2 cannot be an integer > 1

My approach: Since the exact solutions for x and y are almost
impossible one will have to give some convincing arguments.
x and y will probably look like:

x = ( --- + R^1/2)^1/6 and y = (----- +S^1/2)^1/6

Any help will be appreciated



# You appear to have R^(1/2) and S^(1/2) on the
# right, but you say R and S are nonsquare integers.
# Are x and y supposed to be integers? Real?

# Here's a possibly useful identity.

R := (x^6 -21*x^4*y^2 + 35*x^2*y^4 - 7*y^6)^2;

S := (y^6 -21*y^4*x^2 + 35*y^2*x^4 - 7*x^6)^2 ;

factor(x^2*R + y^2*S); # (x^2 + y^2)^7
;quit;
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