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Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permu …

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 …

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 …

To answer your first question: there is indeed no $s$ such that $241+2^{2s+1}$ is a perfect square. -- Proof: $2^{2s+1}$ is always congruent to either $2$, $8$ or $32$ modulo $63$, which makes $241+2^ …

The question itself has already been answered. Let me just add that in case $d = 3$, one can obtain a nice picture by marking all rational points with height less than some upper bound, and projecting …

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And …

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 …

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group generated by the permutation $$ a: \ (m,n) \ \mapsto \ (m-n,m) $$ of order $6$ and the involutions $$ b: \ (m,n) \ \mapsto …

Are there integers $a, b > 1$ such that $\pi = \log_a(b)$? Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$? Note that the transcendence of $\pi$ makes this a problem - otherwise …

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

Assume that there is a big and powerful country with an information-greedy secret service which has backdoors to all internet nodes throughout the world which permit him to observe all exchanged data …

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$ a generalized polynomial if for any distinct integers $m$ and $n$ we have $(m - n)|(f(m)-f(n))$. It is easy to check that polynomial …

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 …

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 …

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)} \ …