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# 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.

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23 views

### 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 ...
70 views

### 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 $x$ (...
1 vote
72 views

1 vote
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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 ...
47 views

### 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 ...
29 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 ...
76 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
37 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 $b-1$. ...
54 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 ...
29 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
51 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 $h_2(k)$ ...
29 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
42 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
56 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 ...
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 ...
919 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 ...
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 $$b_1!b_2! \cdots b_k! < n!.$$
24 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
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 ...
1 vote
221 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
108 views

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. ...
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
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 ...
1 vote
35 views

1 vote
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### 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 ...
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 ...
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 ...
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 $0$. ...
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? ...
81 views

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, ...
1 vote
100 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 ...
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 ...