# Questions tagged [number-theory]

Number theory involves properties and relationships of numbers, primarily positive integers.

378 questions
Filter by
Sorted by
Tagged with
768 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 ...
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 (...
675 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 $n'$ ...
625 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 ...
15k 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 ...
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 ...
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 1st prime, 3 is ...
2k views

(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 ...
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 ...
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 ...
226 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 ...
846 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 ...
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 digits ...
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: ...
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: ...
393 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 ...
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, ...
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....
255 views

### unRSA: solve the private key

Given positive integer n and e, knowing that e<n and that ...
454 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 ...
1k views

### Some Lonely Primes

I know, I know, yet another primes challenge... Related A lonely (or isolated) prime is a prime number p such that p-2, ...
2k views

### Is this a truncated triangular number?

Related OEIS sequence: A008867 Truncated triangular number A common property of triangular numbers is that they can be arranged in a triangle. For instance, take 21 and arrange into a triangle of <...
905 views

### Crazy but Rational Bases

We have many challenges based on base 10, base 2, base 36, or even base -10, but what about all the other rational bases? Task Given an integer in base 10 and a rational base, return the integer in ...
685 views

### Sparse Protractor

Given some positive integer n, design a protractor with the fewest number of marks that lets you measure all angles that are an integral multiple of ...
1k views

### Find the dot product of Rationals

I was at a friend's house for dinner and they suggested the idea of a "Prime-factor vector space". In this space the positive integers are expressed as a vector such that the nth element in the ...
352 views

### Continued Fraction of Digit-wise Sum of Square Roots

Introduction Your task is to generate the first 1000 terms in the continued fraction representation of digit-wise sum of square root of 2 and square root of 3. In other words, produce exactly the ...
1k views

### Regex for multiples of 9

It is easy to describe a finite state machine that recognizes multiples of 9: keep track of the digit sum (mod 9) and add whatever digit is accepted next. Such a FSM has only 9 states, very simple! By ...
138 views

### Find prime factors of sum of non-composite Fibonacci numbers up to n

The Challenge Given a number, find the sum of the non-composite numbers in the Fibonacci sequence up to that number, and find the prime factors of the sum. For example, if you were given 8, the non-...
3k views

### Bertrand's Primes

Bertrand's Postulate states that for every integer n ≥ 1 there is at least one prime p such that n < p ≤ 2n. In order to verify this theorem for n < 4000 ...
731 views

### Verify Cyclic Difference Sets

A cyclic difference set is a set of positive integers with a unique property: Let n be the largest integer in the set. Let r be ...
2k views

### Pascal's Triangle (Sort of)

Most everyone here is familiar with Pascal's Triangle. It's formed by successive rows, where each element is the sum of its two upper-left and upper-right neighbors. Here are the first ...
1k views

### The modulo parity party

You are given an array A of n strictly positive integers, with n ≥ 2. Your task is to map each entry Ai to: 1 if Aj mod Ai is odd for each j such that 1 ≤ j ≤ n and j ≠ i 2 if Aj mod Ai is even for ...
2k views

### Make Zero From First 'n' Numbers

Challenge The challenge is to write a code that takes a positive integer 'n' as an input and displays all the possible ways in which the numbers from 1 - n can be written, with either positive or ...
965 views

### Can square tree rings be generated from primes?

Apparently yes! In three easy steps. Step 1 Let f(n) denote the prime-counting function (number of primes less than or equal to n). Define the integer sequence s(n) as follows. For each positive ...
2k views

### Diluted Integer Sums

A positive integer can be diluted by inserting a 0 between two bits in its binary expansion. This means that an n-bit number has ...
2k views

### Write numbers as a difference of Nth powers

Challenge There are many numbers which can be expressed as the difference of two squares, or as the difference of two cubes, or maybe even higher powers. Talking about squares, there are various ways ...
1k views

### Minimise the count of prime factors through insertion

Given two positive integers A and B, return the position p that minimises the number of prime factors (counting multiplicities) of the resulting integer, when B is inserted in A at p. For example, ...
1k views

### Prime numbers between n and 2n [duplicate]

Bertrand's postulate states that there is always at least 1 prime number between n and 2n for all n greater than 1. Challenge Your task is to take a positive integer n greater than 1 and find all of ...
3k views

### Recursive Collatz Conjecture

The Collatz conjecture postulates that if you take any positive integer, then repeat the following algorithm enough times: ...
2k views

### Find the largest number of distinct integers that sum to n

The Task Given an input positive integer n (from 1 to your language's limit, inclusively), return or output the maximum number of distinct positive integers that ...