# Tagged Questions

Mathematical questions about Algorithms, including the analysis of algorithms, combinatorial algorithms, proofs of correctness, invariants, and semantic analyses. See also (computational-mathematics) and (computational-complexity).

12k views

### How to use the Extended Euclidean Algorithm manually?

I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
6k views

### Efficiently finding two squares which sum to a prime

The web is littered with any number of pages (example) giving an existence and uniqueness proof that a pair of squares can be found summing to primes congruent to 1 mod 4 (and also that there are no ...
12k views

### Simple explanation and examples of the Miller-Rabin Primality Test

Coming from an understanding of Fermat's primality test, I'm looking for a clear explanation of the Miller-Rabin primality test. Specifically: I understand that for some reason, having non-trivial ...
2k views

### A natural number multiplied by some integer results in a number with only ones and zeros

I recently solved a problem, which says that, A positive integer can be multiplied with another integer resulting in a positive integer that is composed only of one and zero as digits. How can ...
3k views

### Is factoring polynomials as hard as factoring integers?

There seems to be a consensus that factorization of integers is hard (in some precise computational sense.) Is it known whether polynomial factorization is computationally easy or hard? One thing I ...
117 views

### How is this possible to convert a long string to a number with less characters?

I'm going to write a program (function) that can convert a long string to a number. For this, first I convert each character (letter) to a number; like ...
18k views

### What algorithm is used by computers to calculate logarithms?

I would like to know how are logarithms calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that ...
19k views

### Solution of tanx = x?

How do I find the solutions of tanx = x upto any number of decimals? (Of course, there is the graphical method but it just helps in finding the approximate value.....
1k views

### Horse Race question: how to find the 3 fastest horses?

There are 25 horses. You can take 5 of the horses at a time and race them. Each horse always finishes the race in the same amount of time, and there are no ties. The only information you get from each ...
3k views

### Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
12k views

### Calculator algorithms

Does there exist a good reference on the algorithms used by calculators, especially on the trigonometric and transcendental functions? I would still like to know how Casio generates its random ...
16k views

### How to solve this recurrence $T(n) = 2T(n/2) + n\log n$

How can I solve the recurrence relation $T(n) = 2T(n/2) + n\log n$? It almost matches the Master Theorem except for the $n\log n$ part.
8k views

### Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
808 views

### Counting subsets containing three consecutive elements (previously Summation over large values of nCr)

Problem: In how many ways can you select at least $3$ items consecutively out of a set of $n ( 3\leqslant n \leqslant10^{15}$) items. Since the answer could be very large, output it modulo $10^{9}+7$. ...
4k views

### How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number?

I am trying to identify the general case algorithm for counting the different ways dice can add to a given number. For instance, there are six ways to roll a seven with two 6-dice. I've spent quite ...
13k views

### How can I find the square root using pen and paper?

Okay, I know this is very basic question. I learned 2 methods in school. But now, I forget one. Here is a simple method that I know. Find the prime divisors of the number Omit the half of numbers ...
931 views

### Proof $\sum\limits_{k=1}^n \binom{n}{k}(-1)^k \log k = \log \log n + \gamma +\frac{\gamma}{\log n} +O\left(\frac1{\log^2 n}\right)$

More precisely, $$\sum_{k=1}^n \binom{n}{k}(-1)^k \log k = \log \log n + \gamma +\frac{\gamma}{\log n} -\frac{\pi^2 + 6 \gamma^2}{12 \log^2 n} +O\left(\frac1{\log ^3 n}\right).$$ This is Theorem 4 ...
4k views

### Are some real numbers “uncomputable”?

Is there an algorithm to calculate any real number. I mean given $a \in \mathbb{R}$ is there an algorithm to calculate $a$ at any degree of accuracy ? I read somewhere (I cannot find the paper) that ...
1k views

### Tranforming 2D outline into 3D plane

I am writing a program where I would like to allow the user to draw 4 connecting lines, such as: And convert this shape into a 3D plane. Is this possible? Is there an existing algorithm to do so? ...
6k views

### How to accurately calculate the error function erf(x) with a computer?

I am looking for an accurate algorithm to calculate the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula 7.1....
4k views

### easy to implement method to fit a power function (regression)

I want to fit to a dataset a power function ($y=Ax^B$). What is the best and easiest method to do this. I need the $A$ and $B$ parameters too. I'm using in general financial data in my project, which ...
903 views

### Algorithms for “solving” $\sqrt{2}$

The very first words out of my mouth need to be this... "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be ...
388 views

### Rewriting repeated integer division with multiplication

In many programming languages, such as C and C++, integer division of positive numbers is defined by simply ignoring the remainder. $5 / 2 == 2$. In general, is it true of positive integers $a$, $b$, ...
7k views

### Fastest way to calculate $e^x$ up to arbitrary number of decimals?

What are other faster methods of calculating $e^x$ up to any number of decimals other than using the Taylor series formula?
2k views

### Detecting perfect squares faster than by extracting square root

Given the radix-$r$ representation of a integer $n$, and a small integer constant $k$, there is an $O(\log n)$ algorithm for detecting whether $n$ is a multiple of $k$, namely, division, which ...
2k views

### Quickest way to determine a polynomial with positive integer coefficients

Suppose that you are given a polynomial $p(x)$ as a black box (i.e. some oracle, to which you feed $x$ and it returns $p(x)$). It is known that the coefficients of $p(x)$ are all positive integers. ...
770 views

### Optimal algorithm for finding the odd spheres

Say we have $N$ [$3 \le N \le 100,000$] spheres indexed as $1,2,3,\cdots N$,all of them have identical weight apart from one.We have to determine which sphere it is (index) by using only the pair of ...
5k views

### Implementation of Monotone Cubic Interpolation

I'm in need to implement Monotone Cubic Interpolation for interpolate a sequence of points. The information I have about the points are x,y and timestamp. I'm much more an IT guy rather than a ...