# Questions tagged [binary]

Questions related with (base 2) representation of numbers and their unique properties arising out of number representation.

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### Generate Max Number of Sequences Separated by Hamming Distance of 3

I'm interested in whether there is an algorithm for generating the maximum possible number of DNA sequences that are $7$ nucleotides long that differ by at least $3$...
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### Efficient computation of number of partitions into powers of 2

Consider the function $a(n)$ defined as the number of partitions of $n$ into powers of $2$. The sequence is given at OEIS. I am trying to calculate $a(n)$ modulo some fixed prime, for large $n$ and ...
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### Generate set of numbers containing 3 consecutive 1, but without the elements of the previous set [closed]

So I have this specific problem that I couldn't figure out. I want to create a set $F_n$ containing all bitstrings that has 3 consecutive 1s, but not those that are already contained in all the ...
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I was reading Gerald Edgar's "Measure, Topology, and Fractal Geometry" when I came across this exercise Let coordinates $(u,v)$ be defined in the plane with origin at one corner of the ...
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### On $(0,1)$-strings and counting

Consider a binary string of length $n$ that starts with a $1$ and ends in a $0$. Clearly there are $2^{n-2}$ such bit strings. I would like to condition these sequences by insisting that the number of ...
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### high school math: summands

Let's say we have a question that asks you to find the amount of all possible integers adding up to a random number, lets just say 1287. However, the possible integers is restricted to explicitly 1's ...
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### Gray code permutation notation

I 'm trying to understand the notation of the Gray code permutation but since I only know 2-row matrix notation for permutations, I would like an explanation for the notations below . I understand the ...
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### 2009th smallest number in base 10 whose binary representation contains even number of 1's

HMMT 2009 Problem 20 : A positive integer is called Jubilant if the number of 1’s in its binary representation is even. For example, $6=110_2$ is a Jubilant number. What is the $2009$th smallest ...
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### Converting a Quartic Term into Quadratic Form in QUBO for Prime Factorization

I'm trying to embed the prime factorization problem into the form of a QUBO. To do so, let $p$ and $q$ be two real positive numbers. We can represent these two numbers as binary numbers, which itself ...
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### Number of binary solutions to an underdetermined linear system

I don't know if the answer to the following question is trivial, to me it's not. Suppose that X is an $m\times n$ matrix, with $n>m$, and assume it is a realization from some continuous ...
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### Can someone please explain the below and if it means that if x=m/2^n and if m is odd then x can be represented in binary in two ways?

Reference image containing the statement Can someone please explain the below text and what it means in simple words? From what I understood it means that if x=m/2^n and if m is odd then x can be ...
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### Finding two' s complement of a fractional binary number

I want to find out 2's complement of binary number (00101101.1100) To find the 2's complement of a binary number, invert all bits (change 0s to 1s and 1s to 0s) and then add 1 to the result. For the ...