MathDB

Define

f : \mathbb{Z}_+ \to \mathbb{Z}_+

such that

f(1) = 1

and

f(n)

is the greatest prime divisor of

n

for

n > 1

. Aino and Väinö play a game, where each player has a pile of stones. On each turn the player to turn with

m

stones in his pile may remove at most

f(m)

stones from the opponent's pile, but must remove at least one stone. (The own pile stays unchanged.) The first player to clear the opponent's pile wins the game. Prove that there exists a positive integer

n

such that Aino loses, when both players play optimally, Aino starts, and initially both players have

n

stones.

Problem Statement