Question 62
a) Let $x$ and $y$ be real numbers such that $x + y$, $x^2 + y^2$ and $x^4 + y^4$ are integers. Prove that the number $x^n + y^n$ is an integer for all $n \in \mathbb{N}$.
b) Find an example of real numbers $x$ and $y$ that are not integers, such that the numbers $x + y$, $x^2 + y^2$ and $x^4 + y^4$ are all integers.
c) Find an example of real numbers $x$ and $y$ that are not integers, such that the numbers $x + y$, $x^2 + y^2$ and $x^3 + y^3$ are integers, but the number $x^4 + y^4$ is not an integer.
(Neven Elezović)
b) Find an example of real numbers $x$ and $y$ that are not integers, such that the numbers $x + y$, $x^2 + y^2$ and $x^4 + y^4$ are all integers.
c) Find an example of real numbers $x$ and $y$ that are not integers, such that the numbers $x + y$, $x^2 + y^2$ and $x^3 + y^3$ are integers, but the number $x^4 + y^4$ is not an integer.
(Neven Elezović)