8

Part of 2024 Middle European Mathematical Olympiad

Problems(1)

Divisibility implies eventually constant

Source: MEMO 2024 T8

k

be a positive integer and

a_1,a_2,\dots

be an infinite sequence of positive integers such that

a_ia_{i+1} \mid k-a_i^2

for all integers

i \ge 1

. Prove that there exists a positive integer

M

a_n=a_{n+1}

for all integers

n \ge M

number theorynumber theory proposedSequenceDivisibilityconstant