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.
237
questions
2
votes
1answer
19 views
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
1answer
73 views
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
1answer
36 views
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 $b-1$.
...
0
votes
1answer
51 views
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 ...
0
votes
0answers
27 views
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 ...
1
vote
1answer
44 views
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 $h_2(k)$ ...
0
votes
0answers
28 views
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 ...
1
vote
0answers
31 views
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 ...
1
vote
0answers
42 views
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 ...
0
votes
1answer
29 views
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 ...
6
votes
2answers
567 views
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 ...
0
votes
2answers
68 views
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 $$b_1!b_2! \cdots b_k! < n!.$$
0
votes
0answers
23 views
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 $x_i$, ...
1
vote
1answer
46 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 ...
1
vote
2answers
116 views
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 ...
1
vote
3answers
106 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 $$\log (...
14
votes
0answers
142 views
What can be proven regarding the differences in power between unary ECMAScript regex functions and primitive recursive functions?
In 2014, inspired by Regex Golf, I started exploring, along with a mathematician going by the name teukon, what could be done in the unary domain in ECMAScript regex that went significantly beyond ...
1
vote
2answers
76 views
Sum duplicates in array problem
I get $n$ numbers, where every number is an integer between 1 and 1000. If within given numbers are duplicates, I sum all of them to one number. I repeat it until I have only distinct elements. As a ...
0
votes
0answers
99 views
Fast factorial computation
I'm trying to solve this problem - https://codeforces.com/problemset/problem/711/E
I've already found and proved that the result is equal to:
$$
1 - \frac{2^n (2^n - 1) \cdots (2 ^ n - k + 1)}{2^{...
0
votes
2answers
138 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.
...
0
votes
1answer
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
1answer
54 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 ...
1
vote
1answer
32 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)}{...
2
votes
2answers
93 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:
$...
1
vote
1answer
17 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 ...
5
votes
2answers
360 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 ...
2
votes
3answers
194 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 ...
0
votes
0answers
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 $0$.
...
0
votes
1answer
58 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?
...
-1
votes
1answer
81 views
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
2answers
140 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 $k$ $(0 \le k \le n \le 500000)$.
...
3
votes
1answer
65 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 ...
1
vote
1answer
58 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....
1
vote
0answers
54 views
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 ...
1
vote
2answers
23 views
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-...
1
vote
0answers
39 views
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 ...
1
vote
0answers
46 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:
...
0
votes
0answers
24 views
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
1answer
96 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
1answer
89 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, ...
1
vote
0answers
97 views
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 ...
0
votes
1answer
49 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 ...
2
votes
0answers
80 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 ...
1
vote
0answers
37 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 ...
3
votes
1answer
116 views
Efficiently prime factorising an integer with an oracle
Suppose you have a program one_factor(N) that, given an n-digit binary number, N, returns ...
1
vote
1answer
72 views
Finding a winner in a "long list"
This is from betting domain which has something that is called a long list: a list of a "home team win/draw/away team win" markets for 13 games.
A punter can select any combination of the possible ...
3
votes
1answer
205 views
Number of ways n can be written as sum of at least two positive integers
I found a solution in Python for this problem, but do not understand it. The problem is how many ways an integer n can be written as the sum of at least two positive integers. For example, take n = 5. ...
2
votes
1answer
54 views
What is the time complexity of determining whether a solution $x$ exists to $x^k \equiv c \pmod{N}$ if we know the factorization of $N$?
Suppose we are given an integer $c$ and positive integers $k, N$, with no further assumptions on relationships between these numbers. We are also given the prime factorization of $N$. These inputs are ...
3
votes
0answers
49 views
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 ...
1
vote
0answers
48 views
Branch and scope globally unique identifiers
Say we are working with a Prolog-like system where variables are dynamically created in different branch contexts and scopes, yet these variables are also globally viewable by the system regardless of ...
