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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

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

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 $\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{$d_n$the binary representation of a number that can't be expressed with a subtraction of two$n$bit strings having a total of$n1'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$bit string with atmost$n1'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$11's$and$n0's\$
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Mallika Prova

Posts: 6
Joined: Thu Dec 05, 2013 7:44 pm