MathDB

Let

q

be a real number with

\frac{1+\sqrt{5}}{2}<q<2

. If a positive integer

n

is represented in binary system as

2^k+a_{k-1}2^{k-1}+\cdots +2a_1+a_0

, where

a_i\in\{0,1\}

, define

p_n=q^k+a_{k-1}q^{k-1}+\cdots +qa_1+a_0.

Prove that there exist infinitely many positive integers

k

with the property that there is no

l\in\mathbb{N}

such that

p_{2k}<p_l< p_{2k+1}

Problem Statement