MathDB

(a) Prove that every positive integer

n

can be written uniquely in the form

n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},

where

k\geq 0

and

0\le m_1<m_2\cdots <m_{2k+1}

are integers. This number

k

is called weight of

n

.
(b) Find (in closed form) the difference between the number of positive integers at most

2^{2017}

with even weight and the number of positive integers at most

2^{2017}

with odd weight.

1