# Questions tagged [integers]

Questions about properties of, working with and algorithms on integers.

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### What is the most efficient way to compute factorials modulo a prime?

Do you know any algorithm that calculates the factorial after modulus efficiently? For example, I want to program: ...
4k views

### Representing Negative and Complex Numbers Using Lambda Calculus

Most tutorials on Lambda Calculus provide example where Positive Integers and Booleans can be represented by Functions. What about -1 and i?
3k views

### Determine missing number in data stream

We receive a stream of $n-1$ pairwise different numbers from the set $\left\{1,\dots,n\right\}$. How can I determine the missing number with an algorithm that reads the stream once and uses a memory ...
532 views

### Overflow safe summation

Suppose I am given $n$ fixed width integers (i.e. they fit in a register of width $w$), $a_1, a_2, \dots a_n$ such that their sum $a_1 + a_2 + \dots + a_n = S$ also fits in a register of width $w$. ...
1k views

### Comparing rational numbers

Given $a,b,c,d \in \mathbb N$ and $b,d \notin \{0\}$, $$\begin{eqnarray*} \frac a b < \frac c d &\iff& ad < cb \end{eqnarray*}$$ My questions are: Given $a,b,c,d$ Assuming we can ...
3k views

### Most efficient algorithm to print 1-100 using a given random number generator

We are given a random number generator RandNum50 which generates a random integer uniformly in the range 1–50. We may use only this random number generator to ...
352 views

### Number of multisets such that each number from 1 to $n$ can be uniquely expressed as a sum of some of the elements of the multiset

My problem. Given $n$, I want to count the number of valid multisets $S$. A multiset $S$ is valid if The sum of the elements of $S$ is $n$, and Every number from $1$ to $n$ can be expressed uniquely ...
1k views

### Language of the values of an affine function

Write $\bar n$ for the decimal expansion of $n$ (with no leading 0). Let $a$ and $b$ be integers, with $a > 0$. Consider the language of the decimal expansions ...
4k views

### What data structure would efficiently store integer ranges?

I need to keep a collection on integers in the range 0 to 65535 so that I can quickly do the following: Insert a new integer Insert a range of contiguous integers Remove an integer Remove all ...
232 views

### Complexity of computing $n^{n^2}$

What is the complexity of computing $n^{n^2},\;n \in \mathbb{N}$?
370 views

### What algorithms exist for solving natural number linear systems?

I'm looking at the following problem: Given $n$-dimensional vectors of natural numbers $v_1, \ldots, v_m$ and some input vector $u$, is $u$ a linear combination of the $v_i$'s with natural number ...
270 views
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### Test if there exists an integer k to add to one sequence to make it a subsequence of another sequence

Suppose that sequence $A$ contains $n$ integers $a_1,a_2,a_3,\ldots,a_n$ and sequence $B$ contains $m$ integers $b_1,b_2,b_3,\ldots,b_m$. We know that $m \geq n$. We assume without loss of generality ...
1k views

### Algorithm for multiplying multivariate polynomials

Let $R$ be a commutative ring. Let $f(x_1, \dots, x_n), g(x_1, \dots, x_n)$ be two multidimensional polynomials in $R$ with maximal total degree $\delta$. How fast can we compute the product of $f$ ...
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### Algorithm to find optimal currency denominations

Mark lives in a tiny country populated by people who tend to over-think things. One day, the king of the country decides to redesign the country's currency to make giving change more efficient. The ...
877 views

### How can I find minimum number required to add to sequence such that their xor becomes zero

Given a sequence of natural numbers, you can add any natural number to any number in the sequence such that their xor becomes zero. My goal is to minimize the sum of added numbers. Consider the ...
559 views

### Detecting overflow in summation

Suppose I am given an array of $n$ fixed width integers (i.e. they fit in a register of width $w$), $a_1, a_2, \dots a_n$. I want to compute the sum $S = a_1 + \ldots + a_n$ on a machine with 2's ...
I'm reading this Intel white paper on carry-less multiplication. It describes multiplication of polynomials in $\text{GF}(2^n)$. On a high level, this is performed in two steps: (1) multiplication of ...