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Let $S \subset \mathbb{R}$ be the set of all real numbers $x$ for which there are infinitely many pairs of prime numbers $p$ and $q$ such that $$ \left|x-\frac{p}{q}\right| < \frac{1}{q^2}. $$ Do th …

Given a polynomial $P$ with integer coefficients in finitely many variables, we denote by $v(P)$ the product of the absolute values of the non-zero coefficients and the non-zero total degrees of the m …

Given a set $A \subseteq \mathbb{N}$ of positive integers, put $$ S_A: \mathbb{N} \rightarrow \mathbb{R}, \ \ \ \ N \mapsto \sum_{n \in A \cap \{1, \dots, N\}} \frac{1}{n}. $$ There are obvious qu …

Given a positive integer $n$ which is not a perfect square, it is well-known that Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$. Question: Let $n$ be a posit …

Is there an upper bound on the number of partitions of a finite set $S$ of prime numbers into 3 sets $A$, $B$ and $C$ for which the following holds?: $$ \prod_{p \in A} p \ + \ \prod_{p \in B} p \ = …

Does the following hold?: $$ \sum_{a, b \in \mathbb{N}^+, \ \gcd(a,b) = 1} \frac{1}{ab(a+b)} \ = \ 2 $$ Numerical computations suggest this may hold, but on the other hand it would be quite surprisi …

Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto \sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective? In 1999, when I was an undergraduate student, I t …

Given an integer $n$, let $P(n)$ denote the set of odd prime divisors of $n$. Let $\Delta$ be the simplicial complex over the set of sets of odd prime numbers which consists of the simplices $S$ such …

Assume that we randomly choose a 100-digit prime number $p$, record which of the first 1000 prime numbers are primitive roots modulo $p$, and then forget about $p$. — How easy or how difficult is it t …

To my knowledge it is open so far whether the polynomial $x^2+1 \in \mathbb{Z}[x]$ takes infinitely many prime numbers as values. Is it known so far whether there is at all any polynomial $P \in \math …

Meanwhile, this question has been answered in the positive. -- See Thomas F. Bloom, Olof Sisask: Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions.

Given a positive integer $n$, let $f_1(n)$ denote the number of pairs of consecutive prime numbers $\leq n$ which are incongruent modulo 3, and let $f_2(n)$ denote the number of pairs of consecutive p …

Question: For which $d, k \in \mathbb{N}$ is there an algorithm to determine whether a polynomial diophantine equation $$ P(x_1, \dots, x_k) = 0, \ \ \ P \in \mathbb{Z}[x_1, \dots, x_k] $$ …

Given positive integers $a$, $m$ and $n$, let $s_{a(m)}(n)$ denote the sum of the reciprocals of the prime numbers less than or equal to $n$ which are congruent to $a$ modulo $m$. Is there an integer …

Let $(a_1,b_1), \dots, (a_k,b_k)$ be finitely many pairs of positive integers, and let $\Gamma$ be the graph whose vertices are the prime numbers and in which two vertices $p$ and $q$ are connected by …