# New algebraic theories?

Number Theory Level 3

I. $$\large x+y = \sqrt{z^2+2xy}$$

II. $$\large x+y = \sqrt[3]{z^3+3xy(x+y)}$$

III. $$\large \left(\dfrac{x}{z}\right)^2+\left(\dfrac{y}{z}\right)^2=1$$

IV. $$\large \left(\dfrac{x}{z}\right)^4+\left(\dfrac{y}{z}\right)^4=1$$

V. $$\large \left(\dfrac{x}{z}\right)^4+\left(\dfrac{y}{z}\right)^4 = \dfrac{z^4-2x^{2}y^{2}}{z^4}$$

Which equation(s) above has/have infinitely many positive integer solutions for $$x$$ , $$y$$ and $$z$$?

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