Questions tagged [number-theory]
number theory is the branch of mathematics concerning the mathematical properties of numbers and the relationships between various types of numbers. This tag should be used with questions regarding computer science topics which are presented from a number theory perspective or may involve number theory or whose answer could be or should be couched in number theory terms.
243
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Period of modulo exponentiation function from factors
The calculation of the period of the "modulo exponentiation" function:
$$
f_a(r) = a^r (mod \ N)
$$
is a step of the quantum algorithm for factorizing $N$, where $a$ is a number chosen with ...
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3
votes
1
answer
70
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Computable Numbers and Cantor's Diagonal Method
We were given the following problem in our university:
We will call $x \in (0; 1)$ computable iff there exists an algorithm (e.g. a programme in Python) which would compute the $n^{th}$ digit of $x$ (...
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1
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2
answers
72
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Why does the hash function in Rabin—Karp produce unique hashes?
The Rabin–Karp algorithm uses a rolling hash function to find a substring matching a pattern:
$$ H = c_1a^{k-1} + c_2 a^{k-2} + c_3 a^{k-3} + \cdots + c_k a^0, $$
where $a$ is a constant, $c_1\ldots ...
- 39
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0
answers
23
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Finding square root of a gram matrix over the integers [closed]
Suppose that matrix A is a symmetric positive definite matrix over the integers, i.e., $A \in Z^{n\times n}$, if B is a matrix over the real numbers, it is not difficult to find B such that $A = B \...
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1
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1
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38
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Where does each part of the $1 - (1 - 1/k)^k$ approximation for the Maximum Coverage problem come from?
A solution to an instance of the Maximum Coverage problem with a budget of k subsets can be approximated with a greedy algorithm that, at each iteration, picks one of the subsets that adds the most ...
2
votes
1
answer
47
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Is there a simplistic way of describing the proof to the undecidability of David Hilbert's 10th problem?
I recently have been reading a bunch about David Hilbert's famous 10th problem, and trying to understand its proof. I am currently in the process of reading through an explanation of the proof, given ...
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1
answer
29
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Why does this algorithm to calculate number of pronic numbers in an interval work?
I am given $A$ and $B$, where $A$ is less than or equal to $B$, and they are in the range of $[1, 100000000]$. I want to calculate the number of pronic numbers in that interval $[A, B]$. These numbers ...
3
votes
1
answer
76
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Collision between a bi-infinite linear sequence of 2D integer lattice points and any of a fixed set of such sequences
Given:
a finite collection $V$ of bi-infinite linear sequences of two-dimensional integer lattice points, each sequence ${V_i}$ given by $\cdots,\vec{{V_i}_{-1}},\vec{{V_i}_0},\vec{{V_i}_1},\cdots$ ...
1
vote
1
answer
37
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algorithm to check the existence of $b-1$ in $(n)_b$?
Given an integer number $n$ (in base 10) and a base $b$, determine whether the representation of $n$ in base $b$, $(n)_b$ has a coefficient with value of $b-1$.
...
- 173
0
votes
1
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54
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How to get the numbers in the Sieve of Eratosthenes with wheel factorisation?
I'd like some help with figuring out the algorithm for the Sieve of Eratosthenes with wheel factorisation. Specifically, I need help figuring out if it's possible to convert between an index and the ...
- 111
0
votes
0
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29
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Number partition subjected to the cardinality of subset
I hope someone can take some time to consider the following problem and welcome to discuss together.
Number partition problem is one of well-known NP-hard problems.
Now I am considering the hardness ...
- 11
1
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1
answer
51
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Why we get at most $N^2$ probe sequences using double hash function
Question:Given the following double hash function:
$$h(k,i) = (h_1(k) + i\times h_2(k)) \bmod{N}$$, where $h_1(k): key \to \mathbb{Z}$. $h(k,i)$ can generate $N^2$ probe sequences at most and $h_2(k)$ ...
- 459
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0
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29
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What is the complexity of (prime?) factorization with a fixed number of primes?
I was wondering what the complexity of factorization (on quantum computers or classical computers) is if we know that there must be exactly two prime numbers and we know the two prime numbers.
For ...
user136613
1
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0
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42
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Computability of base-conversion for streams
With regular numbers it is always possible to convert them from one (integer) base to another. But what happens if we consider numbers for which their magnitude is not known in advance, or in other ...
- 41
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0
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56
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Is it possible to find length of sum of two binary numbers without calculating sum?
I'm doing an assignment where I need to multiply two 16 bit numbers and store result as an 16 bit integer array.
a is binary with length of ...
- 11
0
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1
answer
29
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Sharing a requested secret number not knowing which
Alice has a list of secret numbers. Alice wants to give Bob the opportunity to choose one of her secret numbers, and then Alice has to share that specific secret number, but in such a way that Alice ...
- 165
6
votes
2
answers
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How can we count the number of pairs of coprime integers in an array of integers? (CSES)
For reference, I am trying to solve this CSES Problem.
The problem basically states that given up to $10^5$ positive integers in the range $[1, 10^6]$, find the number of pairs of those positive ...
- 425
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votes
2
answers
68
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Proving $b_1! \cdots b_k! < (b_1 + \cdots + b_k+1)!$ [closed]
I can't figure out how to solve this problem:
Let $b_1,b_2,\ldots,b_k$ be positive integers with sum less than $n$. Prove that $$b_1!b_2! \cdots b_k! < n!.$$
- 17
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0
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24
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Calculating ring inequalities with fixed precision
Suppose we are given $n$ $M$-bit (the word size of the computer) integers $x_i$, and wish to calculate a linear or nonlinear inequality involving the numbers (an expression made up from the $x_i$, ...
- 451
1
vote
1
answer
48
views
An algorithm that finds ord(a) in $O(\log n)$
Let $p$ be a Fermat prime ($p=2^m+1$) and $n=p2^k$ and $a∈Z^*_n$, I should suggest an algorithm that will find $\operatorname{ord}(a)$ in polynomial time (that is, in time polynomial in $\log n$).
I ...
- 117
1
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2
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221
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Half precision floating point question -- smallest non-zero number
There's a floating point question that popped up and I'm confused about the solution. It states that
IEEE 754-2008 introduces half precision, which is a binary
floating-point representation that uses ...
- 13
1
vote
3
answers
108
views
Primes not dividing sequence $a_{n+1} = 1 + a_0 a_1 \cdots a_n$
Prove that there are infinitely many primes that divide none of the elements of the integer sequence
$a_{n+1} =1+a_0 a_1 \cdots a_n$, with a starting point of $a_0 \geq 0$.
I thought about $$\log (...
- 51
14
votes
0
answers
154
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What can be proven regarding the differences in power between unary ECMAScript regex functions and primitive recursive functions?
In 2014, inspired by Regex Golf, I started exploring, along with a mathematician going by the name teukon, what could be done in the unary domain in ECMAScript regex that went significantly beyond ...
- 241
1
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2
answers
83
views
Sum duplicates in array problem
I get $n$ numbers, where every number is an integer between 1 and 1000. If within given numbers are duplicates, I sum all of them to one number. I repeat it until I have only distinct elements. As a ...
0
votes
0
answers
108
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Fast factorial computation
I'm trying to solve this problem - https://codeforces.com/problemset/problem/711/E
I've already found and proved that the result is equal to:
$$
1 - \frac{2^n (2^n - 1) \cdots (2 ^ n - k + 1)}{2^{...
- 101
0
votes
2
answers
142
views
Find the minimum subset of a set of numbers with product divisible by a given integer
The following problem was part of a local programming contest I attended..(I solved it via the obvious Brute Force solution)
I was wondering whether there was a cleaner Dynamic Programming solution.
...
- 13
0
votes
1
answer
60
views
Create an optimal to express all natural number as an arithmetic expression using the alphabet $\Sigma=\{(,),1,+,\times\}$
Hope you had a fantastic christmas break :)
I am trying to find an algorithm in polynomial time that finds the shortest arithmetical expression (the one with the least amount of 1 symbols) to express ...
1
vote
1
answer
74
views
An algorithem for finding the number of primes of the form 4k+3 under some n
I was given the task to make an algorithem that can compute the number of prime's of the form 4k+3 under some n, it should be able to compute how many number's of this type are there under 10^8 (100 ...
- 33
1
vote
1
answer
35
views
Strengthening a given attack on discrete log
I am trying to prove the following claim:
Let $(G,*)$ be a cyclic group of size $m$ with generator $g$. Assume
there exists some adversary $A'$ of size
$T'=\frac{\left(T-O\left(\log m\right)\right)}{...
- 163
2
votes
2
answers
117
views
How to iterate the Hardy-Ramanujan integers quickly
The Hardy-Ramanujan integers, A025487 - OEIS, are integers which when factorized, have their exponents for all the primes starting from 2, in decreasing (not strictly) order. The first few terms are:
$...
- 303
1
vote
1
answer
20
views
Populating a vector of numbers to expose an error in a function implementation
So lets say I'm writing an algorithm that takes a vector as input. I want to know that I'm writing this algorithm correctly however so I of course write tests to see if the output equals what I expect ...
- 3,728
5
votes
2
answers
383
views
Efficient Algorithm to Find the n-th Odious Number
An odious number is defined as an integer that has odd binary Hamming weight. I need an implementation of algorithm that finds the nth odious number, preferably recursive. Any ideas? A python script ...
- 53
2
votes
3
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292
views
Shortest path in modular arithmetic
Suppose we have 7 vertices, each of which corresponds to a different integer modulo seven. The edge exists between two vertices x and y if x + 3 ≡ y mod 7. For example, there is an edge between 0 and ...
- 202
0
votes
0
answers
55
views
Finding index of $p_{k}$ element in the original sorted array if elements were to be removed using a specific condition
Consider a sorted list of numbers $C_{0}=\{0,1,2,3,...,n-1\}$ from where one element will be eliminated at each step. We are also given a value $L$ in $[0, 1)$ and let the indexing start from $0$.
...
- 11
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votes
1
answer
60
views
Given arbitrary integers of $K$ and $M$, can deciding $2^K$ + $M$ is a prime be in $P$?
Given arbitrary integers of $K$ and $M$, is $2^K$ + $M$ a prime?
...
- 309
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votes
1
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81
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Number of sequences of given type
Consider that there is sequence $a$ of length $n$ ,$a=[a_i,0\le i\le n]$. Now you are given with $\text{lcm}$ of some pairs of number from list that is, $\operatorname{lcm}(a_i,a_j)=k$ for $0\le i,j\...
2
votes
2
answers
166
views
Efficient way to reduce a binomial coefficient as a fraction
Here is the full problem.
You need to calculate Euler's totient function of a binomial coefficient $C_n^k$.
Input
The first line contains two integers: $n$ and $k$ $(0 \le k \le n \le 500000)$.
...
- 155
4
votes
1
answer
89
views
Decidability of equality, and soundness of expressions involving elementary arithmetic and exponentials
Let's have expressions that are composed of elements of $\mathbb N$ and a limited set of binary operations {$+,\times,-,/$} and functions {$\exp, \ln$}. The expressions are always well-formed and form ...
- 161
1
vote
1
answer
69
views
Must a decision problem in $NP$ have a complement in $Co-NP$, if I can verify the solutions to in polynomial-time?
Goldbach's Conjecture says every even integer $>$ $2$ can be expressed as the sum of two primes.
Let's say $N$ is our input and its $10$. Which is an integer > 2 and is not odd.
Algorithm
1....
- 57
1
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0
answers
57
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Evaluating functions related by Mobius inversion formula
Problem
Consider two functions $f: \mathbb{N} \rightarrow \mathbb{N}$ and $g: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(k) = \sum_{d | k} g(d)$ for all $k \in \mathbb{N}$. So, the questions ...
- 11
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vote
2
answers
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Relation/use of irrational or transcedental numbers in computer science?
I'm wodering about the relationship between the theory that studies irrational/transcendental numbers and computer science. For example, I found this paper (but was unable to get the full text)
Pseudo-...
- 151
1
vote
0
answers
42
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Decidability of equality of expressions involving exponentiation
Let's have expressions that are finite-sized trees, with elements of $\mathbb N$ as leaf nodes and the operations {$+,\times,-,/$, ^} with their usual semantics as the internal nodes, with the special ...
- 161
1
vote
0
answers
52
views
Rabin Karp algorithm that uses bitwise AND
I'm reading the source code of JPlag and came across their rabin-karp algorithm implemented found here.
Here's the gist of it:
...
- 11
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votes
0
answers
26
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Need help optimizing an algorithm that's supposed to maximize the greatest common divisor of n elements by removing at most one element
Alright, first here's the text of the problem:
You're given n bags of candies where the i-th bag contains a[i] candies and all numbers a[i] are in the segment [1,m]. You can choose a natural ...
1
vote
1
answer
107
views
Finding the smallest number that scales a set of irrational numbers to integers
Say we have a set $S$ of $n$ irrational numbers $\left\{a_1, ..., a_n\right\}$.
Are there any known algorithms that can determine a scaling factor $s \in \mathcal{R}$ such that $s * a_i \in \mathcal{...
3
votes
1
answer
93
views
How to Generate Münchhausen Numbers in High Radices?
A Münchhausen number is a whole number equal to the sum of its digits raised to powers of themselves. For the purpose of such calculations, the convention is that $0^0 = 1$. For example, in radix 10, ...
- 131
1
vote
0
answers
100
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how did the authors of the AKS-Paper come up with the upper bound for r? and what does the multiplicative order have to do with anything?
I have been recently reading the paper "PRIMES is in P", but unfortunately a lot the steps were skipped, which led to confusion. My main problem is with the upper bound on r which was not explained at ...
- 11
0
votes
1
answer
50
views
Given a set of integers $D$ and a positive value$P$, find an algorithm to find set of integers satisfying a condition
Given a set of positive integers :
$ \\ D = \{ D_1, D_2, ..., D_n\}$
and a non-negative integer $P$, where $P$ is divisible by every element in $D$, then find ...
- 115
2
votes
0
answers
81
views
Counting triplets from three arrays satisfying the equation x^2 = yz
Let's say I have three arrays of positive integers X, Y and Z. You can assume that each of ...
- 21
1
vote
0
answers
40
views
Circuit depth of computing the continued fractions of a rational number
If you want to convert a rational number into its continued fraction, what is the circuit depth of this process, in terms of the total number of bits of input?
I was reading through some notes which ...
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