# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
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### What is the smallest integer $n$ for which $\theta(n) > n$?

What is the smallest integer $n$ for which $\theta(n) > n$? Here $\theta(x) = \sum_{p \leq x} \log p$. I googled around, checked some likely textbooks, and ran a program for $n \leq 10^7$, but ...
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### Fastest Primality test using $N-1$ Factorization?

If $N-1$ could be factored easily with several small prime factors, then what is the fastest way to check $N$ for primality? Updated I'm aware of Pocklington primility test which is not good for ...
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### Extensions of Mixed Hodge Structures

The analogy on the front page of this paper by Bloch and Kriz seems like it's going to be lovely, but I don't get it, because I don't know how to view a torsor for $\mathbb{Q}(1)$ as an extension of ...
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### Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p}$ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
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### Sum of the primitive roots

It is well known that the sum of the primitive roots modulo $p$ is congruent to $\mu(p − 1) \bmod{p}$. But I can't see why this result is interesting or useful. Can someone please enlighten me?
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### Proof of infinitude of primes whose reversal in base 10 is also prime

Is there any proof of infinitude of http://oeis.org/A007500 primes. If you want to generate them here is trivial and naive python program. ...
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### Interesting properties of a mathematical number theory game

The game, which is purely recreational, goes as follows: Starting out with 1, you can employ any of two different generation rules: You can multiply by 3 You can divide by two, rounding up (e.g. 3 ...