# Tagged Questions

Prime numbers, diophantine equations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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### Simple proof for $\sum_{i=1}^{n} a^{gcd(i,n)}$ is divisible by n

Burnside's Lemma Deduce That: $$\sum_{i=1}^{n} a^{gcd(i,n)}$$ is divisible by n it's a beautiful result. but i want to prove it without any abstract algebraic tools such as Burnside's Lemma... is ...
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### integral basis of orthogonal complement

Suppose there are $r$ linearly independent vectors $v_1,\dots,v_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v_i\|_\infty\leq m$ for some integer $m$. My goal is to find an ...
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### Chowla's Construction of prime having least quadratic non-residue $\gg \log p$
This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues". I recently ...
### Complexity of $\mathsf{gcd}(a,b)\bmod N$
Given $a,b\in\Bbb N$ where each $a,b$ is $n$-bits, we can compute $\mathsf{gcd}(a,b)$ in $cn^{1+\epsilon}$ bit operations for some fixed $c\geq1$. My query is given $N,a,b$ where $a,b$ is $n$-bits ...