MathDB

Prove that there exist

C>0

, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence

a_1, a_2, a_3,\cdots

, if the greatest common divisor of

a_1

and

a_2

is squarefree, then there exists a positive integer

m\le C\cdot {a_2}^2

, such that

a_m

is squarefree. Note: A positive integer

N

is squarefree if it is not divisible by any square number greater than

1

.
Proposed by Qu Zhenhua

Problem Statement