Questions tagged [number-theory]
Number theory involves properties and relationships of numbers, primarily positive integers.
354
questions
17
votes
16answers
881 views
(RGS 4/5) Inverting matrices modulo m
Task
Given an integer matrix M and a modulus m, find an inverse of M modulo ...
34
votes
45answers
3k views
(RGS 1/5) Binary multiples
A binary multiple of a positive integer k is a positive integer n such that n is written ...
9
votes
5answers
187 views
Count switches in a smallest square root sequence mod \$2^n\$
Definition
For any \$a\equiv1\ (\text{mod }8)\$ and \$n\ge3\$, there are exactly 4 roots to the equation \$x^2\equiv a\ (\text{mod }2^n)\$. Now, let \$x_k(a)\$ be the smallest root to the equation \$...
14
votes
15answers
1k views
Find the Inverse Neighbor Pairs
Definition
We say a pair of integers \$(a,b)\$, where \$0<a<b<N\$ and \$N\$ is an integer larger than 4, is an inverse neighbor pair respect to \$N\$ if \$ab\equiv1\text{ }(\text{mod }N)\$ ...
8
votes
4answers
369 views
Dirichlet Convolution Inverse
If \$f,g\colon \mathbb{Z}_{\geq 1} \to \mathbb{R}\$, the Dirichlet convolution of \$f\$ and \$g\$ is defined by
\$ \qquad\qquad\qquad \displaystyle (f*g)(n) = \sum_{d|n}f(d)g(n/d).\$
This ...
29
votes
5answers
2k views
Longest Prime Sums
Sandbox
There are special sets S of primes such that \$\sum\limits_{p\in S}\frac1{p-1}=1\$. In this challenge, your goal is to find the largest possible set of primes that satisfies this condition.
...
26
votes
13answers
3k views
Who's next to me in the queue?
Problem 4 in the 2019 BMO, Round 1 describes the following setup:
There are \$2019\$ penguins waddling towards their favourite restaurant. As
the penguins arrive, they are handed tickets numbered ...
24
votes
12answers
2k views
Next Shared Totient
The totient function \$\phi(n)\$, also called Euler's totient function, is defined as the number of positive integers \$\le n\$ that are relatively prime to (i.e., do not contain any factor in common ...
28
votes
18answers
4k views
Fermat's Last Theorem, mod n
Fermat's Last Theorem, mod n
It is a well known fact that for all integers \$p>2\$, there exist no integers \$x, y, z>0\$ such that \$x^p+y^p=z^p\$. However, this statement is not true in ...
24
votes
10answers
2k views
Fermat's polygonal number theorem
Fermat's polygonal number theorem states that every positive integer can be expressed as the sum of at most \$n\$ \$n\$-gonal numbers. This means that every positive integer can be expressed as the ...
18
votes
8answers
1k views
Dividing Divisive Divisors
Given a positive integer \$n\$ you can always find a tuple \$(k_1,k_2,...,k_m)\$ of integers \$k_i \geqslant 2\$ such that \$k_1 \cdot k_2 \cdot ... \cdot k_m = n\$ and $$k_1 | k_2 \text{ , } k_2 | ...
12
votes
2answers
326 views
Interpreter for number theory, modulo n
A sentence of number theory (for our purposes) is a sequence of the following symbols:
0 and ' (successor) - successor means <...
13
votes
12answers
436 views
Find all \$k\$-smooth pairs
Introduction
In number theory, we say a number is \$k\$-smooth when its prime factors are all at most \$k\$. For example, 2940 is 7-smooth because \$2940=2^2\cdot3\cdot5\cdot7^2\$.
Here, we define a ...
11
votes
12answers
1k views
Magical Modulo Squares
I'm a big fan of number theory. A big thing in number theory is modular arithmetic; the definition being \$a\equiv b\mod m\$ if and only if \$m\mid a-b\$. A fun thing to do is raising to powers: ...
21
votes
5answers
520 views
Congruent Numbers
Definitions:
A triangle is considered a right triangle if one of the inner angles is exactly 90 degrees.
A number is considered rational if it can be represented by a ratio of integers, i.e., ...
28
votes
26answers
2k views
Fundamental Solution of the Pell Equation
Given some positive integer \$n\$ that is not a square, find the fundamental solution \$(x,y)\$ of the associated Pell equation
$$x^2 - n\cdot y^2 = 1$$
Details
The fundamental \$(x,y)\$ is a pair ...
17
votes
16answers
2k views
Do I have a twin with permutated remainders?
We define \$R_n\$ as the list of remainders of the Euclidean division of \$n\$ by \$2\$, \$3\$, \$5\$ and \$7\$.
Given an integer \$n\ge0\$, you have to figure out if there exists an integer \$0<k&...
29
votes
45answers
9k views
Am I not good enough for you?
Background:
The current Perfect Numbers challenge is rather flawed and complicated, since it asks you to output in a complex format involving the factors of the number. This is a purely decision-...
7
votes
7answers
554 views
Check type of an integer
You will receive an integer less than 2000000000 and bigger than -2000000000 and you have to test what type(s) of number this is out of:
...
12
votes
18answers
844 views
Find a Rocco number
I was asked this question in an interview but I was unable to figure out any solution. I don't know whether the question was right or not. I tried a lot but couldn't reach any solution. Honestly ...
16
votes
23answers
2k views
Find if a list is an ABC-triple
Three positive integers A, B, C are ABC-triple if they are coprime,
with A < B and satisfying the relation : A + B = C
Examples :
1, 8, 9 is an ABC-triple ...
22
votes
13answers
3k views
Eight coins for the fair king
This is a "counterpart" of another puzzle, Eight coins for the fair king on Puzzling.SE.
You can read the above puzzle for the background. The details about this puzzle are as follows.
A set of 8 ...
18
votes
20answers
3k views
Is this number a hill number?
A hill number is a number that has the same digit in the first & the last, but that's not all. In a hill number the first digits are strictly increasing until the largest digit, and after the ...
20
votes
14answers
1k views
Dirichlet Convolution
The Dirichlet convolution is a special kind of convolution that appears as a very useful tool in number theory. It operates on the set of arithmetic functions.
Challenge
Given two arithmetic ...
13
votes
6answers
1k views
Ryley's Theorem
S. Ryley proved following theorem in 1825:
Every rational number can be expressed as a sum of three rational cubes.
Challenge
Given some rational number \$r \in \mathbb Q \$ find three rational ...
15
votes
11answers
900 views
Generate some rough numbers
Background
A number n can be described as B-rough if all of the prime factors of n strictly ...
19
votes
10answers
2k views
Repeat this GCD operation
Problem A3 from the 2008 Putnam competition says:
Start with a finite sequence \$a_1, a_2, \dots, a_n\$ of positive integers. If possible, choose two indices \$j < k\$ such that \$a_j\$ does not ...
18
votes
26answers
3k views
Is the word coprime?
Given a word, treat every letter as it's number in English alphabet (so a becomes 1, b becomes 2, ...
13
votes
10answers
1k views
Quadratic residues are so much fun!
Definitions
Quadratic residues
An integer \$r\$ is called a quadratic residue modulo \$n\$ if there exists an integer \$x\$ such that:
$$x^2\equiv r \pmod n$$
The set of quadratic residues modulo \...
14
votes
2answers
397 views
Impatient divisibility test
Your task is to write a program or function
that determines whether a number is divisible by another.
The catch is that it should give an answer as soon as possible,
even if not all digits of the ...
13
votes
7answers
737 views
Make the biggest and smallest numbers
Inspired by this post over on Puzzling. Spoilers for that puzzle are below.
Given three positive integers as input, (x, y, z), construct the inclusive range ...
15
votes
16answers
2k views
The number of ways a number is a sum of consecutive primes
Given an integer greater than 1, output the number of ways it can be expressed as the sum of one or more consecutive primes.
Order of summands doesn't matter. A sum can consist of a single number (...
12
votes
17answers
635 views
Compute the minimum \$a(n)>a(n-1)\$ such that \$a(1)+a(2)+\dots+a(n)\$ is prime (OEIS A051935)
Background
Consider the following sequence (A051935 in OEIS):
Start with the term \$2\$.
Find the lowest integer \$n\$ greater than \$2\$ such that \$2+n\$ is prime.
Find the lowest integer \$n'\$ ...
17
votes
7answers
620 views
Split the bits!
We define \$V(x)\$ as the list of distinct powers of \$2\$ that sum to \$x\$. For instance, \$V(35)=[32,2,1]\$.
By convention, powers are sorted here from highest to lowest. But it does not affect ...
38
votes
96answers
14k views
Is this number evil?
Introduction
In number theory, a number is considered evil if there are an even number of 1's in its binary representation. In today's challenge, you will be identifying whether or not a given number ...
24
votes
13answers
3k views
Find the 10-adic cube root of 3
I like to think of a 10-adic number as a number that goes infinitely to the left, or an integer modulo a very very large power of 10.
Things carry infinitely to the left and vanish. To see what I ...
22
votes
19answers
2k views
Is this a consecutive-prime/constant-exponent number?
A while ago, I had a look at the prime factorization of 27000:
27000 = 23 × 33 × 53
There are two special things about that:
consecutive-prime: The primes are consecutive: 2 is the ...
27
votes
24answers
2k views
The Add-Multiply-Add Sequence
(Related)
Given an integer n > 1,
1) Construct the range of numbers n, n-1, n-2, ... 3, 2, 1 and calculate the sum
2) Take ...
51
votes
11answers
3k views
Bringing a pair of integers to equality
This was inspired by a math problem I saw somewhere on the internet but do not remember where (UPDATE: The original problem was found on the math riddles subreddit with a proof provided that it is ...
14
votes
15answers
1k views
Am I a Pillai prime?
A Pillai prime is a prime number \$p\$ for which there exists some positive \$m\$ such that \$(m! + 1) \equiv 0 \:(\text{mod } p)\$ and \$p \not\equiv 1\:(\text{mod }m)\$.
In other words, an ...
6
votes
2answers
220 views
Minimal Triangles
Make an upside down triangle of positive integers. Every number in the triangle must be distinct. Each number is the summation of its two parents (similar to how Pascal's triangle is constructed, but ...
22
votes
11answers
796 views
Digit Product Sequences
Here's an interesting sequence discovered by Paul Loomis, a mathematician at Bloomsburg University. From his page on this sequence:
Define
...
5
votes
27answers
2k views
Simple Factorial Challenge [duplicate]
In light of today's date...
A factorial of a number n, is the product of all the numbers from 1 to n inclusive.
The Challenge
Given an integer n where 0 <= n <= 420, find the sum of the ...
18
votes
13answers
2k views
Conway's Prime Game
Specifically, Conway's PRIMEGAME.
This is an algorithm devised by John H. Conway to generate primes using a sequence of 14 rational numbers:
...
14
votes
9answers
1k views
Seidel Triangle
The Seidel Triangle is a mathematical construction similar to Pascal's Triangle, and is known for it's connection to the Bernoulli numbers.
The first few rows are:
...
9
votes
5answers
372 views
Four Spiraling Axes
Take the numbers 0, 1, 2, 3, 4, ... and arrange them in a clockwise spiral, starting downward, writing each digit in its own separate square.
Then, given one of ...
18
votes
26answers
1k views
Rotational Average
Given an input integer n >= 10, output the average of all deduplicated rotations of the integer.
For example, for input 123, ...
32
votes
6answers
2k views
1, 2, 3, 14… or is it 15?
A well known song by the Irish rock band U2 starts with the singer Bono saying "1, 2, 3, 14" in Spanish ("uno, dos, tres, catorce").
There are various theories as to the significance of those numbers....
4
votes
3answers
234 views
19
votes
3answers
400 views
Swap program halves to test divisors
Four integer sequences
In this challenge, you will test four different properties of a positive integer, given by the following sequences.
A positive integer N is
perfect (OEIS A000396), if the sum ...
