MathDB

Alice chooses a prime number

p > 2

and then Bob chooses a positive integer

n_0

. Alice, in the first move, chooses an integer

n_1 > n_0

and calculates the expression

s_1 = n_0^{n_1} + n_1^{n_0}

; then Bob, in the second move, chooses an integer

n_2 > n_1

and calculates the expression

s_2 = n_1^{n_2} + n_2^{n_1}

; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number

n_k

such that

p

divides

s_k(s_1 + 2s_2 + · · · + ks_k)

. Who has a winning strategy?
Proposed by Borche Joshevski, Macedonia

N7