MathDB

Binary representations and sum of the digits

Source: Turkish TST 2011 Problem 6

July 23, 2011

floor functionlimitnumber theorygreatest common divisorinductionprime factorizationnumber theory proposed

Problem Statement

Let

t(n)

be the sum of the digits in the binary representation of a positive integer

n,

and let

k \geq 2

be an integer.
a. Show that there exists a sequence

(a_i)_{i=1}^{\infty}

of integers such that

a_m \geq 3

is an odd integer and

t(a_1a_2 \cdots a_m)=k

for all

m \geq 1.

b. Show that there is an integer

N

such that

t(3 \cdot 5 \cdots (2m+1))>k

for all integers

m \geq N.