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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 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 …

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 …

Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$. What is $$ \limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \ …

Let ${\rm rad}(n)$ denote the radical of a positive integer $n$, i.e. the product of its distinct prime divisors. Given positive integers $a$ and $b$, the triple $(a,b,a+b)$ is called an abc triple if …

The Borel-Cantelli heuristic suggests that for any odd $n \in \mathbb{N}$, there is some $k \in \mathbb{N}$ such that $n+2^k$ is prime -- and for small $n$ this is in fact true (in particular, for any …

Observe that for any $\epsilon > 0$ there are infinitely many triples of $c^\epsilon$-smooth coprime positive integers $a$, $b$ and $c$ such that $a + b = c$. -- Considering triples of the form $(2^n- …

Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number of prime numbers $p \leq n$ in the residue class $r$ (mod $m$). Further let $1 = r_1 < r_2 < \dots < r_{\varphi(m)} = m-1$ …

It seems that a nice example is the set $A$ of positive integers which have an even number of 1's in their binary expansion, although I don't see a reasonable lower bound on $R_k(A)$ for now. A quick …