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.
224
questions
0
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2answers
89 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 $$\log (...
12
votes
0answers
117 views
What can be proven regarding the differences in power between unary ECMAScript regex functions and primitive recursive functions?
In 2014, inspired by Regex Golf, I started exploring, along with a mathematician going by the name teukon, what could be done in the unary domain in ECMAScript regex that went significantly beyond ...
1
vote
2answers
55 views
Sum duplicates in array problem
I get $n$ numbers, where every number is an integer between 1 and 1000. If within given numbers are duplicates, I sum all of them to one number. I repeat it until I have only distinct elements. As a ...
0
votes
0answers
84 views
Fast factorial computation
I'm trying to solve this problem - https://codeforces.com/problemset/problem/711/E
I've already found and proved that the result is equal to:
$$
1 - \frac{2^n (2^n - 1) \cdots (2 ^ n - k + 1)}{2^{...
0
votes
2answers
123 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
44 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
76 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:
$...
0
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0answers
22 views
Radix Economy of Complex Bases
If we extend the allowed bases for a numerical system to the complex numbers, is e still the most economic base? If not, what would it be?
There's the well-known formula for radix economy:
Where b is ...
1
vote
1answer
14 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
295 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
136 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
79 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
110 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
53 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
50 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
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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
35 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 ...
0
votes
0answers
17 views
Which branches of computer science make use of algebraic number theory?
There was similar question about Analytic Number Theory: https://cstheory.stackexchange.com/questions/46555/analytic-number-theory-in-tcs
1
vote
0answers
21 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
20 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
65 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
88 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
95 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
46 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
72 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
29 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
104 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
70 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
94 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
42 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
48 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
47 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 ...
0
votes
0answers
22 views
Prove that x and y in extended Euclid's algorithm won't overflow an Integer (If a,b <= 1e8, ax+by=gcd(a,b))
We are given a and b <= 1e8.
The extended Euclid's algorithm always finds a solution for ax+by=gcd(a,b) (assuming it exists) which can always be stored in an Int.
How to prove the x and y won't ...
0
votes
1answer
46 views
Prove, a^2+b^2=c^2,there exists only 1 case such that a,b,c are consecutive non negative integers(3,4,5) [closed]
I want to prove, $a^2+b^2=c^2$,there exists only 1 case such that a,b,c are consecutive non-negative integers(3,4,5).
I have no clue to prove this lemma. Please help me to prove this lemma.
5
votes
0answers
86 views
Given $n=pq=a^2+b^2$, can we factor $n$?
Just to be clear, $a$ and $b$ are known, while $p$ and $q$ are unknown prime numbers, both congruent to $1$ modulo $4$. Can we design an efficient algorithm to retrieve $p$ and $q$?
It is a known ...
0
votes
1answer
83 views
Guess the number from its different base representations
Given a set of numbers in different representations (we don't know the value of the base in which we are representing) of bases, find the original number (in decimal representation) if it exists or ...
0
votes
1answer
102 views
How to Optimise the following algorithm? maximum good value of an element in an array
I was recently stuck while doing a question, please suggest a way with a code/pseudo code to optimize the following algorithm for finding the maximum good value in an array where a good value of an ...
1
vote
1answer
32 views
Simple generator of pseudo-random permutations of variable length short sequence
The problem in front of me is to write a function (from scratch) to permute n elements, where n is an argument. I decided to break it down to applying Knuth's shuffles algorithm, therefore I needed to ...
1
vote
1answer
25 views
Use Fact to decompose a given number
Let Fact be the following decision problem. Given two natural numbers $k \le n$ the machine accepts if and only if $n$ has a non-trivial divisor that is less than or equal to $k$. Prove that if Fact $\...
0
votes
1answer
70 views
How to count all integers less than a given integer and having two contigous digits as $y$?
Suppose i have been given a number 54432 .How to count all numbers less than 54432 and having last two digits as 1 ? i.e all the numbers of form xxx11 and xxx11 < 54432 .Here x can be any digits ...
2
votes
1answer
53 views
How do I minimize the cost of some algorithm that performs some operation on a list?
I stumbled upon this problem whilst studying the complexity of a simple algorithm. I used set-theoretic notation, but all the $S_i$'s are lists (I couldn't think of a better way to write the problem ...
0
votes
1answer
66 views
Express a number with a combination of arithmetic operations and a single number
Today I ran into a this relatively simple (At least from my perspective ) problem. Basically the task is to be able to express any number using only a single number an any combination of arithmetic ...
3
votes
2answers
96 views
Proving that a set of operations can't generate one integer from a given one
Given two numbers, $n$ and $m$, are there some mathematical methods of deducing $m$ from $n$ using limited number of elemantary operations?
Example: 335 can be deduced from 2000 using division by 2, ...
3
votes
4answers
212 views
How to select best k fractions out of n fractions (k<=n) so as to have (numerator sum / denominator sum) maximum?
For example, given 4 fractions $\frac{4}{2}$, $\frac{2}{3}$, $\frac{1}{2}$, $\frac{10}{20}$, I have to select 3 fractions out of these 4 so that the value of $\frac{\text{numerator sum}}{\text{...
-1
votes
1answer
123 views
I want to find the number of steps it takes to find the GCD by Euclidean Algorithm
Let's say I have two numbers a and b. I want to find the number of steps it takes to find the GCD by Euclidean Algorithm by a closed formula which includes parameter a and b.
If I go by this ...
