Question 60
Let $M$ consist of integers of the form $a^2+13b^2$, where $a,b$ are nonzero integers.
- Prove that the product of any two elements of $M$ belongs to $M$.
- Determine, with justification, whether there are infinitely many integer pairs $(x,y)$ such that $x+y$ does not belong to $M$ but $x^{13}+y^{13}$ does.
- Prove that the product of any two elements of $M$ belongs to $M$.
- Determine, with justification, whether there are infinitely many integer pairs $(x,y)$ such that $x+y$ does not belong to $M$ but $x^{13}+y^{13}$ does.