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# Questions tagged [number-theory]

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

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14 votes
5 answers
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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 $2^k \times 2^k$ ...
• 3,638
16 votes
6 answers
1k views

### 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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13 votes
11 answers
804 views

### *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 (...
• 78.2k
10 votes
4 answers
2k views

### 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 ...
• 2,123
21 votes
15 answers
2k views

### 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 ...
• 2,123
15 votes
16 answers
1k views

### 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 ...
• 487
18 votes
26 answers
2k views

### 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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11 votes
10 answers
1k views

• 2,279
16 votes
2 answers
607 views

• 10.6k
11 votes
12 answers
1k views

• 50.1k
16 votes
15 answers
1k views

### 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 ...
• 513
15 votes
2 answers
1k views

### 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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