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Let $K$ be a number field. $O_K$ be its ring of integers, so $O_K^*$ are the units. We have sequence $1 \rightarrow O_K^* \rightarrow K^* \rightarrow K^*/O_K^* \rightarrow 1$ Note that $K^*/O_K^*$ is …

Let $a, b, n$ be positive integers. Assume that $\gcd(a,b,n)=1$. It seems that one can prove that there exist two integers $c$ and $d$ bounded from above by $( \log n )^{O(1)}$ such that $ \gcd (ac …

Is there a good asymptotic estimation for the number of integral points inside a $n$-sphere of a small radius ( less than $\sqrt{n}$)? It looks like a problem which has been studied, however, I cannot …

Let $p$ be a prime. Let $F_p$ be the finite field of $p$ elements. Let $A$ be a subset of $F_p$ of size $s$. Assume that $s > 2$ is polylogarithmic in $p$. Suppose that we want to count number o …

Let $p$ be a prime. Given a positive integer $n$, does there exist a $\beta$ in an extension of $F_p$ such that 1) If $F_p[\beta] = F_{p^N}$, then $N > n^n$; ( $\beta$ lies in a high extension) 2) …

Let $q$ be a power of 2. Let $P$ be the set of polynomials in $F_q [x]$ of degree d or less. Let $\mathbb{Z}$ be the ring of integers. For any $f \in P$, let $\psi(f)$ be the number of distinct roo …