# Questions tagged [number-theory]

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

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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Digit Product Sequences

Here's an interesting sequence discovered by Paul Loomis, a mathematician at Bloomsburg University. From his page on this sequence: Define ...
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### 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 ...
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### 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: ...
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### 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: ...
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### 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 ...
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### Rotational Average

Given an input integer n >= 10, output the average of all deduplicated rotations of the integer. For example, for input 123, ...
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### 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....
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### unRSA: solve the private key

Given positive integer n and e, knowing that e<n and that ...
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### 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 ...
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### 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, ...
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### 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 <...