MathDB

Nondivisible power sums

Source: Kyiv City MO 2024 Round 2, Problem 9.2

February 4, 2024

number theory

Problem Statement

You are given a positive integer

n > 1

. What is the largest possible number of integers that can be chosen from the set

\{1, 2, 3, \ldots, 2^n\}

so that for any two different chosen integers

a, b

, the value

a^k + b^k

is not divisible by

2^n

for any positive integer

k

?
Proposed by Oleksii Masalitin