MathDB

Let

a_1<a_2<...<a_t

t

given positive integers where no three form an arithmetic progression. For

k=t,t+1,...

define

a_{k+1}

to be the smallest positive integer larger than

a_k

satisfying the condition that no three of

a_1,a_2,...,a_{k+1}

form an arithmetic progression. For any

x\in\mathbb{R}^+

define

A(x)

to be the number of terms in

\{a_i\}_{i\ge 1}

that are at most

x

. Show that there exist

c>1

and

K>0

such that

A(x)\ge c\sqrt{x}

for any

x>K

Problem Statement