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Anthony Quas has already answered the first part. The second supremum is $1$ as well -- just let $A$ be the set of positive integers all of whose prime divisors are $2$ or $\equiv 1 (\!\!\mod 4)$, and …

Actually for all $11 \leq k \leq 100$ which satisfy the conditions in the question (i.e. $\gcd(k,105) = 1$ and $\mu(k) \neq 0$), the coefficient of $x^7$ in $\Phi_{105k}(x)$ equals $0$. Hence so far t …

Is there a set $A$ of positive integers such that $\sum_{n \in A} \frac{1}{n} = \infty$, and there is no polynomial $f \in \mathbb{Z}[x]$ of degree at least $2$ which takes infinitely many values in …

The values $n(p)$ for primes $13 \leq p < 100$, found by computation: $n(13) = n(17) = 716$, $n(19) = 62987$, $n(23) = 367082$, $n(29) = 728366$, $n(31) = 64822396$, $n(37) = 1306238012$, $n(41) = 111 …

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.

Let $m$ and $n$ be distinct odd positive integers. The equality $$ \prod_{k=0}^{mn-1} \left( e^{\frac{2\pi i k}{m}} + e^{\frac{2\pi i k}{n}} \right) \ = \ 2^{\gcd(m,n)} $ …

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 …

There is no such $m_0$, due to the Erdős–Kac theorem.

Carl Pomerance conjectured in On the Distribution of Pseudoprimes, Math. Comput. 37, 587-593 (1981) that for large $x$, the number of pseudoprimes $\leq x$ is $$ \frac{x}{e^{(1+o(1))\log{x}\frac{\ …

Yes, there is precisely one such triple of distinct primes with $\alpha \leq 500$, namely $(3, 11, 31)$. It would be a surprising coincidence if there are further such triples for larger $\alpha$.

We have the following polynomial identity: $$ 3^{12k+6} - b^{24\ell} \ = \ (3^{3k+2} b^{2\ell} - b^{8\ell})^3 + (3^{4k+2} - 3^{k+1} b^{6\ell})^3. $$ Therefore Question 1 …

Given a prime number $p$ and a primitive root $a$ modulo $p$, let $\sigma_{a,p}$ denote the permutation of the set $\{1, \dots, p-1\}$ which maps $b$ to $a^b$ modulo $p$. Question: Let $p$ be fixed. …

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 …

For abelian simple groups your question is merely a disguised form of "enumerate the Mersenne primes". The smallest examples for nonabelian simple groups are as follows: $|{\rm PSL}(2,7)| + 1 = 13^2 …

If the sequence $(T(N))_{N \in \mathbb{N}}$ converges and the limit is not equal to $0$, this would imply either positive predecessor density for $1$, cf. e.g. Günther J. Wirsching, On the problem of …