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.
173 questions
2
votes
1answer
50 views
+50
How does the bitlength of the divisor affect the running-time complexity of division algorithms?
Wikipedia lists $O(M(n))$ as the best complexity (out of the algorithms listed) for division on two $n$-digit numbers, where $M(n)$ is the complexity of the multiplication algorithm of choice. This is ...
1
vote
2answers
28 views
Running time complexity of finding maximal power of divisor that divides natural number
Given $n \in \mathbb{N}$, a divisor $p\vert n$, I would like to efficiently find $e\in\mathbb{N}$ with $p^e \vert n$, and $e$ maximal with this property. I will assume that multiplication/division of ...
2
votes
1answer
39 views
Generalizing Knuth's $O(\log_2 n)$ Fibonacci algorithm to linear homogenous recurrences
Knuth has a neat algorithm that uses matrix exponentiation to compute the $n$th Fibonacci number in $O(\log_2 n)$-time 1. However, there doesn't seem to be a lot of resources on generalizing his idea ...
2
votes
1answer
34 views
Bertrand's ballot theorem
I want to understand the dynamic programming equation of https://en.wikipedia.org/wiki/Bertrand%27s_ballot_theorem theorem.
it is this
If i number of people voted for A and j number of people voted ...
1
vote
1answer
37 views
Carpet into Box
Given a carpet of size a * b [length * breadth] and a box of size c * d, one has to fit the carpet in the box in the minimum number of moves. A move is to fold the carpet in half, either by length or ...
2
votes
2answers
27 views
An approximate quantity of multiplications in $\mathbb{F}_p$ amounting the same bit complexity as one inversion in $\mathbb{F}_p$
Consider a prime finite field $\mathbb{F}_p$ of quite large characteristics $p$, for example $\log_2(p) \approx 256$ bits. I would like to know an approximate quantity of multiplications in $\mathbb{F}...
-2
votes
1answer
47 views
Number Theory Problem from Local Selection Contest EPFL | ETHZ
This was a question from the 2016 local (selection) contest in ETHZ, You have a high-precision alarm clock with three operations:
1) reset wake-up time to midnight (00:00:00.000000)
2) modify the ...
2
votes
2answers
57 views
Algorithm for generate all solutions to a linear Diophantine equation
Consider the linear Diophantine equation of the form:
$$\sum_{i=1}^{k}a_ix_i=n.$$
My goal is to list all the non-negative solutions to this equation. I wrote the following recursive algorithm, but I ...
1
vote
1answer
45 views
Modular exponentiation in P
I need to prove that the following language is in $\mathsf{P}$:
$$ L = \{\langle x,y,z,d \rangle : xˆy \equiv z \pmod{d}\}.$$
I'm assuming I just have to prove with an algorithm or negate that it's ...
2
votes
0answers
19 views
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 ...
1
vote
1answer
92 views
Checking Divisibility Using Minimal Bits
Suppose we're given an infinite stream of integers, $x_1, x_2, \dots$.
a) Show that we can compute whether the sum of all integers seen so far is divisible by some fixed integer $N$ using $O(\log ...
1
vote
2answers
115 views
Randomly Choosing a N-Bit Prime
I've been studying some number theory, and I came across this problem:
Lagrange’s prime number theorem states that as N increases, the number of primes less than $N$ is $Θ(N/ log(N))$.
...
1
vote
1answer
28 views
How to calculate $\sum_{i=1}^n \mu^2(i)$ in less than $O(n)$'s time
To go with $O(n)$, we can use the linear sieve according to that $\mu(n)$ is multiplicative. But it seems that we don't have to work each $\mu(n)$ out and accumulate them together, because I only want ...
3
votes
0answers
14 views
Base-k representations of polynomials: state of art [closed]
In chapter 4 of Jeffrey Shallit's A Second Course in Automata Theory the following problem is formulated as open:
Let $p(n)$ be a polynomial with rational coefficients such that $p(n) \in \mathbb{N}$ ...
1
vote
1answer
28 views
Two's complement max with a different base
Working with fixed size integer representations,
use a number system with b-complement notation, base b = 9 and
n = 4 digits.
What is the smallest number that can be represented in this number system?...
1
vote
3answers
66 views
Determining number of elements divisible by at least one prime of set
I have an array ($|A|\leq 10^6$) of numbers ($A_i\leq10^6$) and a set of prime numbers. I have to find the count of the elements in the array that are divisible by at least one of the numbers in the ...
2
votes
1answer
57 views
How to estimate the average time complexity of greatest common divisor?
As we know, the time complexity of $\gcd(x,y)$ is $O(\log \min(x,y))$ by using Euclidean algorithm. Now we fix a constant $n$ and consider the average time complexity of $\gcd(x,n)$.
Formally, let $f(...
0
votes
1answer
25 views
Maximum trailing zeros of the path
Problems:
A table with $n$ rows and $m$ columns is filled with number from $1$ to $100$ (duplication allowed). The player starts at $(1, 1)$. He can only move right or down. The goal is to reach $(n, ...
3
votes
2answers
65 views
Count numbers less than $x$ co-prime to $p$
We have given two numbers $x$ and $p$. We want to count how many numbers are less than $x$ and are co-prime with $p$.
I know that we can solve the problem in $O(x\log x)$ with iterating over all ...
4
votes
2answers
117 views
Which is the fastest method for calculating exact square root of a integer of 200-500 digit number?
I wanted to know is there any algorithm / function / process through which I can calculate square root of a very large integer number. I wants to know current state of the research in this field.
No ...
0
votes
1answer
73 views
What is the most compact (space efficient) method of storing an array of distinct integers?
I have an array of distinct integers which I want to save in the most compact manner. I may have to do occasional lookups, deletions, and insertions in this array so the compression algorithm must ...
4
votes
0answers
69 views
Optimal parallel-time repeated modular squaring circuit
Given a 4096-bit integer $x$ and a 4096-bit RSA modulus $N$ (of unknown factorisation) what is the fastest circuit to compute $x^{2^T} \mod N$ where $T=2^{40}$. That is, what is the fastest parallel-...
0
votes
1answer
30 views
How hexadecimal representation is more compact and intelligible for documentation?
My textbook says,
"Instead, it is far better to use a hexadecimal representation for documentation purposes. Whether or not a code represents a binary number, it can be treated as ...
6
votes
0answers
33 views
Level sums, displacements: how to determine their effect efficiently?
Let $R =\mathbb{Z}/N \mathbb{Z}$. Let $f:R\to \mathbb{R}$,
$\rho:R\to \lbrack 0,1\rbrack$. We assume that it takes trivial time to compute any given value $f(m)$ or $\rho(m)$.
Define $$S(\delta,m) = ...
6
votes
0answers
161 views
Testing algorithm for a modified sieve of Eratosthenes
Context:
I am looking at a modified version of the sieve of Eratosthenes. I started by generalising Eratosthenes' sieve, like so:
Choose some starting "root", $n_0\in\mathbb{N}$, a sieve limit (the ...
4
votes
2answers
80 views
Proving that $\{0^{m^2}\mid m\geq 3\}^*$ is regular
We know that $L=\{0^{m^2}\mid m\geq 3 \}$ is not a regular language. However $L^*$ is regular because we can generate $0^{120}$ to $0^{128}$ by some concatenations and then any other power of $0$ can ...
-2
votes
1answer
70 views
Symmetric difference of a set with an empty set [closed]
The definition of symmetric difference of two sets $\alpha $ and $\beta$, $\alpha
\oplus \beta$ is defined as the set of all $x$ such that, $x \in (\alpha \cup \beta) - (\alpha \cap \beta)$.
If, $\...
0
votes
1answer
6 views
Systematically altering functions
Let A and B be two integers, where their size is constrained to 2 bits in binary. There exists a function F, which outputs integers of the same size, where F(A) = F(B) = Y. For example, A = 01, B = 10,...
2
votes
1answer
915 views
Confusion in 2's complement of 00000000
I'm solving the end of the chapter problems of Morris Mano's Digital Design (4th Edition, if that's relevant). In one of the problems, it is asked to simply find the 1's and 2's complement of 00000000....
2
votes
2answers
133 views
Pumping lemma: the set of strings of 0s and 1s such that when interpreted as an integer, that integer is prime
In the section of my textbook covering the pumping lemma, there are practice questions asking us to prove a given language is not regular.
I have not been able to solve this one:
The set of ...
1
vote
0answers
39 views
System of congruences with non-pairwise coprime moduli
I have a set of congruences
x ≡ a1 (mod n)
...
x ≡ ak (mod nk)
And I want to find x, this can be solved by the Chinese ...
2
votes
3answers
109 views
Closest divisors in array
Given an array $A[1],\ldots,A[n]$ of natural numbers, we have to construct a new array $B[1],\ldots,B[n]$, where $B[i]$ is equal to $A[j]$ for the minimal $j > i$ such that $A[j]$ divides $A[i]$, ...
2
votes
2answers
175 views
How can I find a number whose sum of digits' cube is equal to n
I'm given a number $n$ and I have to find the smallest number possible whose sum of digits' cube is equal to $n$. For example $n=9$ then output should be $12$, because $1^3 + 2^3=9$.
1
vote
1answer
41 views
Is there any bitwise multiplication algorithm that is sub O(n^2)?
The following program implements a simple algorithm for binary multiplication:
...
0
votes
1answer
111 views
Find n-th number in a number system with only 3 and 4
Given a number system with only 3 and 4. Find the nth number in the number system.
There is a solution given here but its very vague without explaining the mathematics/ concept behind it. Can someone ...
0
votes
1answer
22 views
Higher Residuocity Problem w.r.t. a composite Modulus
An integer $q$ is called a quadratic residue modulo $N$ if it is congruent to a perfect square modulo $N$; i.e., if there exists an integer $x$ such that:
$x^2≡q\ (mod\ n)$.
Otherwise, $q$ is called ...
3
votes
1answer
87 views
Calculating $\sum_{i=1}^a \lfloor a/i \rfloor i^2$
I have a sum:
$$S = \sum_{i=1}^n{\lfloor a/i \rfloor i^2},$$
where $a$ is a constant.
Is there a way to speed this up? That is, can we avoid iterating overl all $i$s, possibly calculating it in ...
3
votes
0answers
34 views
How hard is it to find the length of multiplicative 2-partition?
Some terminology at first:
Multiplicative 2-partition for number $N$ - a pair of numbers $\{A, B\}$ such that $AB=N$.
Minimal multiplicative 2-partition length (denoted $l$) - minimal total number of ...
3
votes
1answer
359 views
Computation of discrete logarithm
I know that an equation
$$a^x \equiv b \pmod{p}$$
can be solved for $x$ in $O(\sqrt{p} \ log(p) )$ time using meet-in-the-middle technique, which relies on fact that we can rewrite the equation as ...
1
vote
0answers
80 views
Algorythm for creating Number-Rows
Given is a list of numbers.
Now you build different permutations of that list while there must not be two permutations where the sum of the numbers from any point of the row to the end/beginning is ...
-1
votes
1answer
741 views
Calculating mod of n^k mod p in O(log k)
Given $n>0$, $k>0$, and a prime number $p$, compute: $$n^k\mod p$$ in $O(\log k)$ time. Here $x \mod y$ means the remainder for $x$ being divided by $y$. For example, $2^3\mod 7 = 1$. You assume ...
1
vote
0answers
50 views
How does this last step in Shor's algorithm work?
Page 854 of The Nature of Computation states the following (This discussion has made the simplifying assumption that $M/r$ is an integer):
If each of the observations gives us a random harmonic, ...
1
vote
1answer
46 views
Why do Alice and Bob get the same key even after modular reduction in Diffie-Hellman Cryptography?
In this cryptography scheme, we take $g$ and $p$ (a large prime number). and taking the customary clients Alice and Bob who chose $a$ and $b$ secretly whereas $p$ and $g$ are known in public. I haven'...
1
vote
0answers
37 views
eVoting with Damgard-Jurik-Cryptosystem
I am trying to implement a secure elecetronic voting system. Therefore I found the Damgard and Jurik Cryptosystem. In their paper the authors describe a secure protocol for "A Length-Flexible ...
1
vote
1answer
28 views
Generate a unique number for a set of sequences of letters
A sequence of English letters are given, every sequence forms a word and we know that the size of each word is at most $15$ and the number of words is at most $50000$. For a given input word $w$ I ...
0
votes
2answers
82 views
Find the first integer with exactly $n$ divisors
For given positive integer $n$ find the first number $x$ with exactly $n$ divisors. For simplicity of the problem we can assume that $x$ will always fit in integer size and $n \leq 1000$.
Example
...
1
vote
0answers
33 views
Reducing integer-rooted polynomial to natural-rooted polynomials
First of all, let me define $Dioph(M)$, where $M$ is a set of numbers:
$$Dioph({M})=\{p \mid \text{ $p$ is a polynomial with integer coefficients and the zeros of $p$ are in ${M}$}\}.$$
Let $p \in ...
5
votes
1answer
137 views
Can the semantics of a numeric hierarchy be faithfully represented in Haskell?
I am trying to represent a fragment of a number hierarchy using the Haskell concepts of value, type, and type class. I would like the Haskell code to reflect the mathematical semantics $\vdash ((x \in ...
1
vote
1answer
56 views
Finding solutions to $n m\equiv x \mod n$
Given a modulus $n\in\mathbb{N}$ and another natural number $0<x<n$, what's an efficient algorithm to enumerate all pairs of natural numbers $(a,b)\in\mathbb{N}^2$ such that both $a$ and $b$ are ...
1
vote
1answer
118 views
Is there an online algorithm for radix conversion?
Suppose I have $P$, which is the base-$p$ representation of an integer $n$ and I want to calculate it base-$q$ representation $Q$. The obvious algorithm is: interpret $P$ to obtain $n$, then ...
