N9

Part of 2021-IMOC

Problems(1)

Finite set ensures every number written as sum of powers

Source: IMOC 2021 N9

Find all pairs of positive integers

(a,b)

such that there exists a finite set

S

satisfying that any positive integer can be written in the form

n = x^a + y^b + s

x,y

are nonnegative integers and

s \in S