[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include(/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php) [function.include]: failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 122: include() [function.include]: Failed opening '/home/shoeb/public_html/www.matholympiad.org.bd/forum/includes/phpbb-latex.php' for inclusion (include_path='.:/opt/php53/lib/php')
[phpBB Debug] PHP Warning: in file [ROOT]/includes/session.php on line 1042: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4786: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4788: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4789: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
[phpBB Debug] PHP Warning: in file [ROOT]/includes/functions.php on line 4790: Cannot modify header information - headers already sent by (output started at [ROOT]/includes/functions.php:3887)
BdMO Online Forum • View topic - Numbers expressible as $\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}}$

Numbers expressible as $\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}}$

For discussing Olympiad Level Number Theory problems
Facebook Twitter

Numbers expressible as $\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}}$

Post Number:#1  Unread postby rah4927 » Sun Feb 19, 2017 11:15 am

Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form \[\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},\]
where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$
rah4927
 
Posts: 108
Joined: Sat Feb 07, 2015 9:47 pm

Re: Numbers expressible as $\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{

Post Number:#2  Unread postby Mallika Prova » Thu Apr 13, 2017 1:20 pm

$d_n$ the binary representation of a number that can't be expressed with a subtraction of two $n$ bit strings

having a total of $n$ $1's$ together.

we will prove that for a $d_n < 2^n $ there exists such $n$ bit strings

for a number $x < 2^n$ is a $n$ bit string with atmost $n$ $1's$

now for $a_0$ adjacent $0's$ we can find $\underbrace{10101010...1010}_{a_0}$ and $\underbrace{10101010...1010}_{a_0}$

and for $a_1$ adjacent $1's$ we can find $\underbrace{11111111...1111}_{a_1}$ and $\underbrace{00000000...0000}_{a_1}$


and we are done.. having $d_n=2^n$ a binary string with $1$ $1's$ and $n$ $0's$
If you do what interests you , atleast one person is pleased.
User avatar
Mallika Prova
 
Posts: 6
Joined: Thu Dec 05, 2013 7:44 pm
Location: Mymensingh,Bangladesh


Share with your friends: Facebook Twitter

  • Similar topics
    Replies
    Views
    Author

Return to Number Theory

Who is online

Users browsing this forum: No registered users and 1 guest