Questions tagged [number-theory]
Number theory involves properties and relationships of numbers, primarily positive integers.
471
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Sum-of-four-squares grid
Output a grid of characters visualizing the decomposition of a number into a sum of four perfect squares.
Challenge
Given a nonnegative integer \$0 \le n \le 2^{30}\$, output a \$2^k \times 2^k\$ ...
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votes
6
answers
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Golfing the complexity with subtraction
The Mahler-Popken complexity, \$C(N)\$, of a positive integer, \$N\$, is the smallest number of ones (\$1\$) that can be used to form \$N\$ in a mathematical expression using only the integer* \$1\$ ...
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votes
11
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*Trivial* near-repdigit perfect powers
Task
Output the sequence that precisely consists of the following integers in increasing order:
the 2nd and higher powers of 10 (\$10^i\$ where \$i \ge 2\$),
the squares of powers of 10 times 2 or 3 (...
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votes
4
answers
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Output a 1-2-3-5-7... sequence
Follow-up of my previous challenge, inspired by @emanresu A's question, and proven possible by @att (Mathematica solution linked)
For the purposes of this challenge, a 1-2-3-5-7... sequence is an ...
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votes
15
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Output a 1-2-3 sequence
For the purposes of this challenge, a 1-2-3 sequence is an infinite sequence of increasing positive integers such that for any positive integer \$n\$, exactly one of \$n, 2n,\$ and \$3n\$ appears in ...
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votes
16
answers
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Pretty Palintiples
Imagine you have a positive integer number \$n\$. Let \$m\$ be the number obtained by reversing \$n\$'s digits. If \$m\$ is a whole multiple of \$n\$, then \$n\$ is said to be a reverse divisible ...
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votes
26
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Is it a tetrate of two?
The tetration operation consists of repeated exponentiation, and it is written ↑↑. For instance,
3↑↑3 =3 ^(3^3) = 3^27 = 7,625,597,484,987
A tetrate of two is an ...
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Egyptian fraction representations of 1 without prime denominators
Background
As noted in this question, for all positive integers \$n>2\$ there exists at least one Egyptian fraction representation (EFR) of \$n\$ distinct positive integers \$a_{1} < a_{2} < \...
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votes
5
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Generate a sequence of \$n\$ consecutive composite numbers
Definitions
The common methods to generate consecutive composites are
$$\overbrace{(n+1)! + 2, \ (n+1)! + 3, \ \ldots, \ (n+1)! + (n+1)}^{\text{n composites}}$$
$$\overbrace{n!+2,n!+3,...,n!+n}^{\text{...
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20
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Modular Equivalence
Given two numbers \$x,y > 2, x≠y \$ output all integers \$m\$ such that
$$
x + y \equiv x \cdot y \pmod m
$$
$$
x \cdot y > m > 2
$$
Input
Two integers
Output
A list of integers
Test cases
<...
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votes
10
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Make 1's and 2's composite
Input
An integer k composed of 1 and 2, with at least 3 digits and at most 200 digits.
...
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3
votes
28
answers
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Consecutive Composite Numbers
Challenge
Generate \$n-1\$ consecutive composite numbers using this prime gap formula
$$n!+2,n!+3,...,n!+n$$
Input
An integer \$n\$ such that \$3 \leq n \leq 50 \$.
Output
Sequence of \$n-1\$ ...
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17
votes
19
answers
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Ellipse Lattice Point Counter
Challenge
Determine how many integer lattice points there are in an ellipse
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1$$
centered at the origin with width \$2a\$ and height \$2b\$ where integers \$a, ...
- 2,279
16
votes
2
answers
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Construct this point
Given a constructible point \$(x, y) \in \mathbb R^2\$, output the steps required to construct \$(x, y)\$
Constructing a point
Consider the following "construction" of a point \$(\alpha, \...
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3
votes
2
answers
350
views
Visualise the Euclidean GCD [duplicate]
The Euclidean GCD Algorithm is an algorithm that efficiently computes the GCD of two positive integers, by repeatedly subtracting the smaller number from the larger number until they become equal. It ...
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5
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Random factorized numbers
Input
The code should take an integer \$n\$ between 1 and 1000.
Output
The code should output positive integers with \$n\$ bits. Accompanying each integer should be its full factorization. Each ...
- 3,153
6
votes
4
answers
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Sums of Euler's totient function in sublinear time
Related.
Given a number \$n\$, Euler's totient function, \$\varphi(n)\$ is the number of integers up to \$n\$ which are coprime to \$n\$. That is, no number bigger than \$1\$ divides both of them.
For ...
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votes
11
answers
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Sums of sum of divisors in sublinear time
Given a number \$n\$, we have its sum of divisors, \$\sigma(n)\ = \sum_{d | n} {d}\$, that is, the sum of all numbers which divide \$n\$ (including \$1\$ and \$n\$). For example, \$\sigma(28) = 1 + 2 +...
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11
votes
12
answers
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The all-high powerful numbers
We've had powerful numbers, yes, but what about highly powerful numbers?
Highly powerful numbers
Let \$n\$ be a positive integer in the form
$$n = p_1^{e_{p_1}(n)}p_2^{e_{p_2}(n)}\cdots p_k^{e_{p_k}(n)...
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23
votes
31
answers
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Is this a powerful number?
A powerful number is a positive integer \$n\$ such that for every prime \$p\$ that divides \$n\$, \$p^2\$ also divides \$n\$. Or equivalently, \$n\$ is powerful if and only if it can be written in the ...
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vote
0
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Print all semimagic squares [closed]
I am working on a code to print all semimagic squares [1] of a given size. I am working with the following definition:
An \$n\times n\$ consists of numbers \$1,2,\cdots, n^2\$.
All numbers must be ...
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19
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Sophie Safe primes
Description
Write a program or function that takes in a positive integer \$n\$ as input and outputs all Sophie Germain primes that are safe primes less than or equal to \$n\$. A prime number \$p\$ is ...
- 1,802
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16
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Imtiaz Germain Primes
Description
"Imtiaz Germain primes" is not a technical name in Mathematics, but my weird creation, in the memoir of the famous mathematician Sophie Germain. These primes can be generated by ...
- 1,802
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votes
24
answers
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Shortest code to generate all Pythagorean triples up to a given limit
Generate the shortest possible code in any programming language that can generate all Pythagorean triples with all values not exceeding a given integer limit. A Pythagorean triple is a set of three ...
- 1,802
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votes
25
answers
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Find the Prime Signature
The Prime Signature of a number is the list of the exponents of the prime factors of a number, sorted in descending order (exponents of 0 are ignored). Inspired by ...
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votes
9
answers
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Implement the Riemann R function
The Riemann R function is as follows:
$$R (x)=\sum _{n=1}^{\infty } \frac{\mu (n) \text{li}\left(x^{1/n}\right)}{n}.$$
This uses the Möbius function as well as the logarithmic integral.
From Wikipedia,...
- 3,153
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votes
4
answers
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Write a number as a sum of Fibonacci numbers
In 2009, Hannah Alpert described the "far-difference" representation, a novel way of representing integers as sums and differences of Fibonacci numbers according to the following rules:
...
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"Prime" pyramid
The pyramid begins with the row 1 1. We'll call this row 1. For each subsequent row, start with the previous row and insert the current row number between every ...
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votes
22
answers
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Even and Odd kinds
Let \$n\$ be some positive integer. We say that \$n\$ is of even kind if the prime factorisation of \$n\$ (counting duplicates) has an even number of integers. For example, \$6 = 2 \times 3\$ is of ...
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Advanced Binary Number System
Your task is to write a program that calculates the amount of different ways to display any given whole positive number using the following rules:
Meet the 'advanced binary system':
Any whole positive ...
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10
answers
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IMO Question Six with a difference
In 1988, the International Mathematical Olympiad (IMO) featured this as its final question, Question Six:
Let \$a\$ and \$b\$ be positive integers such that \$ab + 1\$ divides \$a^2 + b^2\$. Show ...
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14
answers
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The "Fly straight, dammit" sequence
Background
"Fly straight, dammit" (OEIS A133058) is a sequence of integers, which has these rules:
\$a_0 = a_1 = 1\$
\$a_n = a_{n-1}+n+1\$ if \$gcd(a_{n-1}, n) = 1\$
Otherwise, \$a_n = \...
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14
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Persistence of a number
The persistence of a number \$x = d_1d_2d_3...d_n\$, with \$d_1 \ne 0\$, under some function \$f : \mathbb N_0 \times \mathbb N_0 \to \mathbb N_0\$ is defined as the number of applications of \$f\$ to ...
- 50.1k
10
votes
18
answers
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Sum of partition numbers
The partition function:
In number theory, the partition function p(n) represents the number of possible partitions of a positive integer n into positive integers
For instance, p(4) = 5 because the ...
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votes
22
answers
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The second even sublime number
easy mode of my previous challenge
A perfect number is a positive integer whose sum of divisors (except itself) is equal to itself. E.g. 6 (1 + 2 + 3 = 6) and 28 (1 + 2 + 4 + 7 + 14 = 28) are perfect.
...
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17
votes
8
answers
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An algorithm to find even sublime numbers
A perfect number is a positive integer whose sum of divisors (except itself) is equal to itself. E.g. 6 (1 + 2 + 3 = 6) and 28 (1 + 2 + 4 + 7 + 14 = 28) are perfect.
A sublime number (OEIS A081357) is ...
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5
votes
2
answers
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Generate a Kirkman triple system
Given a universe of \$v\$ elements, a Kirkman triple system is a set of \$(v-1)/2\$ classes each having \$v/3\$ blocks each having three elements, so that
every pair of elements appears in exactly ...
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Anti-divisors of a number
Given a positive integer n, output all of its anti-divisors in any order.
From OEIS A006272:
Anti-divisors are the numbers that do not divide a number by the ...
- 78.2k
2
votes
0
answers
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Decompose number N into the sum of three triangular numbers [closed]
It is known that any natural number can be decomposed into the sum of three triangular numbers (assuming 0 is triangular), according to Fermat's Polygonal Number Theorem. Your task is to come up with ...
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Help me design an unfair laundry machine
There's a payment machine for laundry in my building which does a few frustrating things. The ones relevant to this challenge are:
It doesn't make change. So if you pay over the amount then you are ...
- 99.6k
3
votes
11
answers
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Divide by an odd number, 2-adically
Given \$a\$ and \$b\$, both odd \$n+1\$-bit integers, compute \$a/b\$ to a precision of \$n+1\$ bits in the 2-adic integers. That is, compute \$c\$ such that \$a = bc\, (\mathop{\rm mod} 2^{n+1})\$. \$...
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Consecutive coin flips
This is a cross-post of a problem I posted to anarchy golf: http://golf.shinh.org/p.rb?tails
Given two integers \$ n \$ and \$ k \$ \$ (0 \le k \le n) \$, count the number of combinations of \$ n \$ ...
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votes
13
answers
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Count alternating permutations
An alternating permutation is a permutation of the first \$ n \$ integers \$ \{ 1 ... n \} \$, such that adjacent pairs of values in the permutation alternate between increasing and decreasing (or ...
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votes
19
answers
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What's the best die to roll?
The "standard" polyhedral game dice have 4, 6, 8, 10, 12 and 20 sides. (Yes, I know that there are two 10-sided dice which together make a d100, but we're ignoring that right now.)
If I want ...
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votes
12
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Calculate the Lowest Even-Harmonic of the Values in a List
PROBLEM
For a list of numbers, list: Find the lowest possible integer, x, which is optimally close to the whole number even-...
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votes
12
answers
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In between fractions
Given two positive integer fractions \$x\$ and \$y\$ such that \$x < y\$, give the fraction \$z\$ with the smallest positive integer denominator such that it is between \$x\$ and \$y\$.
For example ...
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vote
1
answer
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Best performance on x/(y+z) + y/(x+z) + z/(x+y) = N
Consider the equation $$\frac x {y+z} + \frac y {x+z} + \frac z {x+y} = n$$ for positive integers \$x, y, z\$ and \$n \ge 4\$. Your code will receive \$n\$ as an input, and output three integers \$x, ...
- 1,548
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votes
17
answers
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Join my exclusive friendly club!
Two or more positive integers are said to be "friendly" if they have the same "abundancy". The abundancy of an positive integer \$n\$ is defined as $$\frac {\sigma(n)} n,$$ where \$...
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votes
15
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Average-ignorant sets of integers
Recently a friend of mine posed the following:
What subset of the integers satisfies the condition if distinct a and b are in the subset, their average is not in the subset? I know the set of non-0 ...
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votes
2
answers
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Find the magic numbers to divide a number without division
An integer \$x\in[0,2^{32}-1]\$ divided by an integer \$d\in{[1,2^{31}]}\$ will produce an integral quotient \$q\$ and a remainder \$r\$, so that \$x=d\times q+r\$.
Any \$q\$, in fact, can be ...
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