# Questions tagged [number-theory]

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

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### Restricted-source, take this!

a.k.a. You Can Output Anything With Labyrinth Or Hexagony™ Challenge In a recent restricted-source challenge, I could print any character with only half of the allowed digits with very small character ...
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### Narcissistic loop lengths

A narcissistic number is a natural number which is equal to the sum of its digits when each digit is taken to the power of the number digits. For example $8208 = 8^4 + 2^4 + 0^4 + 8^4$, so is ...
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### Repetend length in 1/n

This problem is based on non-terminating, repeating decimal points. Let $n$ be any positive integer $(n > 1 \text{ and } n < 10000)$, say $7$. Then, $1/n = 1/7 = 0.142857142857142857...$ ...
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### Count the Collatz survivors mod 2^n

Introduction We have 22 Collatz conjecture-related challenges as of October 2020, but none of which cares about the restrictions on counter-examples, if any exists, to the conjecture. Considering a ...
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### Find a divisibility pattern

Background Sometimes when I'm golfing a program, I'm presented with the following situation: I have an integer value $x$ on some fixed interval $[a, b]$, and I'd like to test whether it's in some ...
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### 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 $ab\equiv1\text{ }(\text{mod }N)$ ...
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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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### 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 | ...
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### 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 <...