# Questions tagged [integers]

Questions about properties of, working with and algorithms on integers.

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### What are applications to sort plain integer arrays?

A lot of research and engineering effort is put into finding fast methods to sort an array of integers; e.g., Java's runtime library has highly-tuned methods to sort arrays of each primitive type (see ...
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### Best complexity of parity/comparison in the Residue Number System

Let: $\left\{m_1, ~...~, m_k\right\}$ be a set of coprime natural numbers, $M=\prod_{i=1}^{k} m_i$ $X$ be a natural integer, such that $X < M$ Then $X$ can be expressed in the Residue Number ...
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### Quick calculation for $x^y \bmod 2^d$

I need to calculate $x^y \bmod 2^d$ in $O(d)$ summations/bitwise operations and $1$ multiplication by $y$. $x$ is restricted to be odd, $d\geq 3$. $a$-bit arithmetic (for any $a$) is allowed, as this ...
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### String to small integer mapping without collision

Is there any good approach to devise a mapping of limited number of strings $N_1 << 2^{15}$ to integers less than $2^{15}$ without conflicts? Strings are quite often of the form of prefix + ...
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### For two large (10^6) integers A and B, find the number of set bits in the result of their multiplication

I've recently stumbled upon a problem that I struggle realy hard with ever since. There are two large decimal integers, order of magnitude $10^6$. The task is to find how many bits in the binary ...
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### Is finding a sum of two squares representation for a number computationally assymptotically equivalent to integer factorization?

I have been wondering this question because given a number $n$ with prime factors of the form $4k+3$ when we square $n$ and find a sum of two squares which should reveal these types of factors (as ...
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### Are there any practical drop-in replacements for BSTs in the case where data are integers?

There are a number of specialized data structures that implement ordered dictionaries for integer keys: van Emde Boas trees, y-Fast tries, fusion trees, etc. Each of these data structures implement ...
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### How does prefix summing the count histogram in counting sort result in an array of output indices?

My implementation of counting sort is based on the description I found here. I've also included my code in this REPL, but here it is: ...
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### Given $n$ sets of matrices, find $n$ matrices that have the least number of LCDs among their entries

Let's say I have $n$ sets of matrices  A = \left\{\begin{pmatrix} 2 & 4 & 17\\ 5 & 6 & 9\\ \end{pmatrix} \begin{pmatrix} 2 & 4 & 18\\ 5 & 6 & 9\\ \end{pmatrix} \right\...
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### Efficient triangular decomposition of an integer

Euclidean division is an iterative process that has been made super-efficient at the CPU level, right? Its specification is that if I do (q, r) = f(n, d), I get ...
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### What is the currently know fastest algorithms to convert each individual Java primitive and Number types to a Char Array/String

What is the currently know fastest algorithms to convert each individual Java primitive and Number types to a Char Array/String number representation while ...
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### How computer understand to either represent the original bits or two's complement as value?

Important note I know that this question may seem too simple for you scientists; however, here is the best place that I know to post it. Question Suppose we have an ...