Question 114
Aline and Théo play the following game. First, Théo chooses prime numbers $p_{1}, p_{2}, \ldots$ Then Aline chooses pairwise distinct nonzero natural numbers $n_{1}, n_{2}, \ldots$ Théo then chooses two integers $k \geqslant 1$ and $\ell \geqslant 1$. He wins the game if there are infinitely many integers $n_{i}$ that are not divisible by $p_{k}^{\ell}$. Otherwise, Aline wins.
a) Which of Aline and Théo has a winning strategy?
b) To make the game even harder, Aline now has an additional constraint: each integer $n_{i}$ may divide only finitely many integers $n_{j}$. With these new rules, which of Aline and Théo has a winning strategy?
a) Which of Aline and Théo has a winning strategy?
b) To make the game even harder, Aline now has an additional constraint: each integer $n_{i}$ may divide only finitely many integers $n_{j}$. With these new rules, which of Aline and Théo has a winning strategy?