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3
votes
0answers
61 views
+100

How to compute the sum of this series involving golden ratio, efficiently?

Definitions Let $\tau$ be a function on natural numbers defined as $\tau(n)=\lceil n*\phi^2\rceil$ where $n$ is some natural number and $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio. This series ...
2
votes
1answer
51 views

Why is the set of perfect squares in P?

I am reading an article by Cook [1]. In it he writes: The set of perfect squares is in P, since Newton's method can be used to efficiently approximate square roots. I can see how to use Newton's ...
0
votes
1answer
21 views

Computing losing positions in modified Wythoff's game efficiently

Wythoff's game is as follows: there are two players $A$ and $B$ ( $A$ being the first player ) and there are $2$ piles of stones. When his turn a player can remove one or more stones from anyone ...
2
votes
1answer
60 views

Reccurrence for the game of pile of stones

I am trying to solve this question from Project Euler for past few days: Divisor game. The problem is as follows: Two players are playing a game. There are $k$ piles of stones. When it is his turn ...
1
vote
1answer
18 views

Convert integer of mixed radix to standard positional numeral system and vice versa

I have multiple numbers (e.g. [1, 4, 2]) where each number can be one of a specified range of numbers (e.g. [0-1, 0-5, 0-3]). I ...
3
votes
0answers
9 views

Numerical Stability of Halley's Recurrence for Integer $n^{\mathrm{th}}$-Root

tl;dr? See last paragraph. If I use the initial value $2^{\left(\big\lfloor\lfloor\log_2 x \rfloor/n\big\rfloor + 1\right)}$ with Halley's recurrence in the compact form $ x_{k+1} = ...
18
votes
2answers
333 views

How to find the element of the Digit Sum sequence efficiently?

Just out of interest I tried to solve a problem from "Recent" category of Project Euler ( Digit Sum sequence ). But I am unable to think of a way to solve the problem efficiently. The problem is as ...
5
votes
1answer
99 views

Sum of divisors summatory function with Erathosthenes' sieve

I ran across the following problem from an online problem bank: there are up to $~10^5~$ queries each of which asks to compute the sum $$\sum_{k = L}^{R} \sigma(k)$$ where $\sigma(k)$ is the sum of ...
2
votes
1answer
40 views

How does RAID-5 algorithm locate the right device?

Please consider the following diagram of a RAID-5 array (Ignore the gray background): Now, given a logical address, how can one return the device number (0-3)? For example, ...
2
votes
1answer
22 views

First Harshad number with given sum

How can we compute the least Harshad number with given sum of decimal digits $S$? Harshad number in base 10 is any number divisible by sum of its decimal digits. I think that some kind of dynamic ...
0
votes
1answer
45 views

How to compute Jacobi symbol efficiently?

How do I compute the Jacobi symbol $(N|A)$ efficiently? In particular, for every odd $N, A$, define the Jacobi symbol $(A|N)$ as $\prod_i Q_{p_i}(A)$ where $p_1, \dots , p_k$ are all the (not ...
1
vote
1answer
25 views

Calculating product of totient functions

Suppose that i need to compute product $~\phi(1) \dots \phi(n)~$ modulo prime $500009$ as fast as possible. Memory limitations are rather tough: $64$ mb so i can't just keep all the values of totient ...
1
vote
1answer
20 views

Sieving to compute first values of totient function

What i want to do is to compute the first $n$ values of Euler's totient function effectively. As far as i know the best way to do it is to run a sieving algorithm and for each prime encountered to ...
1
vote
1answer
56 views

What is more efficient: gcd(x,y) or brute force, when x and y are a very big numbers

I implemented the quadratic sieve algorithm as it's described in wiki. Most of the work of the algorithm is to determine if some big integer $Y$ belongs to the vector $b[b_1,b_2,b_3,\ldots]$. So far ...
2
votes
2answers
38 views

Shortest path in divisors graph

There is a graph with N vertices numbered from 1 to N. Edge between a and b exists if and only if a|b or b|a. If a|b then the weight of the edge is b/a. If b|a then the weight of the edge is a/b. ...
6
votes
2answers
29 views

Is the following intuition valid for understanding $k$-wise independent hash functions?

I really enjoy reading up on algorithms and data structures and in the course of doing so often come across ones that rely on $k$-wise independent families of hash functions. I'm perfectly comfortable ...
3
votes
1answer
111 views

Find the longest contiguous subsequence such that its sum $(a_i + a_{i+1} + \cdots + a_j)$ is divisible by $D$

You are given $N$ $(1 \le N \le 10^6)$ positive integers $a_1, a_2, \ldots, a_N (1 \le a_i \le 10^6)$ and two positive integers $D$ $(1 \le D \le 10^6)$ and $M$ $(1 \le M \le 10^6)$ Find the longest ...
8
votes
2answers
131 views

Is there any efficient algorithm for primality testing for numbers that are of the form $4k+3$ using the square root function?

I was reading CLRS and it asked to show that if $p$ is a prime of the form $4k+3$ and $a$ was a quadratic residue, then $a^{k+1}$ is a square root (one can also easily show that $a^{-k}$ is a square ...
4
votes
1answer
35 views

How does one find a non-quadratic residue modulo $p$?

I was wondering how one can find a non-quadratic residue modulo $p$ and what the runtime of this algorithm would be. I thought that one can use the Legendre Symbol $$ \left( \frac{a}{p} \right) = ...
2
votes
2answers
149 views

Number of special paths between two nodes

Consider a directed graph. Each node in this graph has an integer label. We want to count the number of special paths between source and ...
2
votes
3answers
257 views

Counting an integer's divisors without just enumerating them (or estimating if not possible)?

I'm trying to count the number of divisors an integer $n$ has. The simple way to do this is to just enumerate all integers from 1 to $\sqrt{n}$, and count how many integers evenly divide $n$. For ...
2
votes
0answers
101 views

Determining whether a number is a perfect square without computing its square root

One of the interesting results of Number Theory is the theory of quadratic reciprocity. One finds that it is possible to determine whether an equation $x^2 \equiv a \pmod p$ has a solution $x$ without ...
2
votes
1answer
88 views

How many operations at maximum needed to compute powering of a number, i.e. $N^k$?

Suppose $N$ is $L$-bit long and $k$ is also $L$-bit long. How to show that it takes at most O$(L^2)$ operations to compute it. For example $N = 1010$ and $k = 10$. Then $N^k = 6^2 = 100100$. But I ...
-1
votes
2answers
444 views

number of subsets where GCD equals to X

The original statement for this problem can be found here This is a question from IEEExtream 2014. There is an array of integers given. Input is X, so output is the number of subsets where there GCD ...
4
votes
1answer
81 views

Least Common Non-Divisor

Basically, the problem is: For a set $S$ of positive numbers, find a minimal number $d$ that is not a divisor of any element of $S$, i.e. $\forall x \in S,\ d \nmid x$. Denote $n = |S|$ and $C = ...
5
votes
4answers
189 views

Checking whether a number divides any of a set of other numbers

Consider a static set $S$ of positive numbers from range $[1..10^9]$. Is there an efficient algorithm to answer queries of the form: "Given a number $d$ is there an element $x \in S$ such that $d | ...
2
votes
1answer
64 views

Number of ways an integer $n$ can be split into a product of $k$ distinct integers given a prime factorization

What is the best algorithm for computing the number of ways an integer $n$ can be split into a product of $k$ distinct integers given a prime factorization $n=p_1^{a_1}p_2^{a_2} \ldots p_i^{a_i}$? ...
3
votes
1answer
90 views

Determining if (infinite) binary language DFAs contain at least 1 prime?

This problem has been given by Shallit as an open DFA/ complexity theory problem and is currently not even known to be decidable. It seems to be circulating on the internet in a few places (e.g. ...
5
votes
2answers
91 views

Finding the k-th smallest rational number efficiently

Consider the following set: $S := \left\{\frac{a}{b} \colon a \in \{1,\ldots,A\}, b \in \{1,\ldots,B\} \right\}$ $S$ is the set of all rational numbers that can be represented by two integers $a$ ...
3
votes
1answer
83 views

Devising an Algorithm for Linear Combination with Column Restrictions

Application: We intend to factor an integer $N$ using a variation of the rational sieve. This involves constructing a congruence of squares modulo $N$ from a set of linear relations $$x - N = y$$ ...
-1
votes
1answer
26 views

How n (1+b) is not prime? [closed]

Here is the complete proof taken from this link How do I convince myself that n(1+b) is not prime when b>=1? Here is what I did: if n is 3 and b is 3. Then ...
3
votes
0answers
50 views

How is Chinese Remainder Theorem used in the proof of correctness for RSA

Question At the very end of (most) proofs of RSA's correctness we have something like $$m^{ed}\equiv m\pmod p$$ $$m^{ed}\equiv m\pmod q$$ Therefore by the Chinese Remainder Theorem (CRT) ...
1
vote
1answer
56 views

Quadratic Diophantine equation - Polynomial Time Cases

In number theory, solving a Quadratic Diophantine equation (a, b, c constants) $$ a*x^2+b*y= c $$ is an NP-Complete problem. Even for a=1, the problem remains NP-Complete. The solution (x, y) are ...
2
votes
1answer
40 views

Efficient computation of traces of all primitive elements in field extensions of GF(2)

Let $n > 1$ and let $\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. The trace function $T(x) = x + x^2 + x^{2^2} + \cdots + x^{2^{n-1}}$ is an onto linear transformation from ...
1
vote
0answers
22 views

Finding all “smooth” numbers up to 10^6 — problem with nested loops [duplicate]

A [smooth number] is one whose factors are all "small". For a computational problem, I would like all smooth numbers up to $10^8$. I think it's unfeasible to factor a billion numbers. Instead, I ...
6
votes
3answers
280 views

How can I generate first n elements of the sequence 3^i * 5^j * 7^k?

How can I efficiently generate the first N elements of the sequence $3^i 5^j 7^k$, where $i,j,k \in \mathbb{N}$? I've googled around a bit and found the sequence in OEIS, but I don't really see a ...
1
vote
1answer
58 views

Introduction to number theory [closed]

What is the best book for a beginner in Introduction to number theory? I am new to this field and getting deeper into cryptography, so I think reading some intro books about number theory can be of ...
-1
votes
2answers
118 views

What is complexity of checking whether a natural number is a perfect square? [closed]

As the title says, what is complexity of checking whether a natural number is a perfect square?
1
vote
1answer
50 views

Smallest possible integer not obtained from sumset

Given a number N, and some set $A=\{a, 1\le a\le N\}$, and let $B=\{\text{every integer} \in [1,N]\}$, and $C=B\setminus A$ (Set C has all values from B not in A) What is the best way of finding the ...
5
votes
1answer
50 views

Fast polynomial calculation over $\mathbb{Z}_{487}$

Given a polynomial $a(x)$ of degree at most $242$ over $\mathbb{Z}_{487}$, I'd like to choose distinct values $x_0, x_1, . . . , x_{242} ∈ \mathbb{Z}_{487}$, such that I'll be able to calculate $a(x_j ...
2
votes
2answers
57 views

Finding the values of x and y using base 8 [closed]

In finding the values of x and y, if (x567) + (2yx5) = (71yx) ( all in base 8) I proceeded as under. I assumed x=abc and ...
3
votes
1answer
104 views

Listing integers as the sum of three squares $m = x^2 + y^2 + z^2$

I would like to write an algorithm that lists all the ways a natural number $m \in \mathbb{N}$ can be written as the sum of three squares $m = x^2 + y^2 + z^2$ with integers $x,y,z \in \mathbb{Z}$. ...
5
votes
1answer
932 views

What does the set {n | n is an integer and n = n + 1} represent?

I am reading Michael Sipser's book Introduction to the Theory of Computation, which mentions the set $$S = \{ n \mid \text{$n$ is an integer and $n = n + 1$}\}.$$ This doesn't make any sense to me. I ...
6
votes
1answer
68 views

Runtime of Euclidean Algorithm

Given two $n$-bits numbers $a$ and $b$, I am not sure on how to find the runtime of the euclidean algorithm for finding the $\gcd$ of $a,b$. The problem (for me) in here is that apart from the size of ...
1
vote
1answer
28 views

A problem regarding Extended Euclidean Algorithm

Alex has some (say, n) marbles (small glass balls) and he has going to buy some boxes to store them. The boxes are of two types: Type 1: each box costs c1 Taka and can hold exactly n1 marbles Type ...
4
votes
1answer
68 views

Fix two primes p and q. Given input a number of the form $p^aq^b$, find the immediate next number of the same form

As stated in the title: Fix two primes p and q. Given input a number of the form $p^aq^b$, find the immediate next number of the same form. For example: when $p = 2$ and $q = 3$ Next of $2^2*3=12$ is ...
1
vote
2answers
107 views

Proving that $2^n$ does not divide $n!$ [closed]

How can prove that $2^n \nmid n!$ using binary representation for $n!$ and $2^n$.
0
votes
2answers
52 views

Algorithm for generation of Canonical represented integers [closed]

Is there a way to get canonical representation of an integer by algorithm? I can get primes, but I don't know how to represent, for example, 1000 as 2^3×5^3. How ...
5
votes
1answer
136 views

Complexity of Pythagorean triples

We define a Pythagorean triple as a triple $\langle a,b,c\rangle$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $\langle a,b,c\rangle$ is legit ...
1
vote
1answer
42 views

Number of nonnegative solutions of linear Diophantine inequality

Given inequality $Ax + By \le C$, where $A, B, C$ are integers, $A$ and $B$ are coprime and $C < AB$. I need to find number of non-negative integer solutions of it. Is there exists algorithm which ...