9 problems

Let $a,m\in\mathbb{N}$ satisfy … Let $\operatorname{GEM}(a,m)$ be the class of polynomials introduced in the paper, and let $\operatorname{GEM}_1(f;a,m)$ and…

Infinitely many zero-Pisano-Legendre primes conjecture. If $0\in\chi(a,b)$, then

Let $a$ and $b$ be integers with $(a,b)\neq(0,0)$, and let $\chi(a,b)$ be the set of integers $k$ for which a $k$-Pisano-Legendre prime relative to $(a,b)$ exists. Infinitude conje…

Quadratic lower-bound conjecture. For every such $a$ and $b$, there exists a real number $c$ with $0<c<1$ such that

Let $p^{(k)}_n$ denote the $k$-th prime in the subsequence of primes having prime index, with $n$ specifying the subsequence, and let $p^{(k)}_k$ be the corresponding diagonal term…

Let $(r_k(n))_{k\geq 0}$ be the companion sequence associated with the positive integer parameter $n$, and let $\mathcal{T}_p(j)$ denote the relevant dilated Chebyshev value. Prime…

Let $(s_k(n))_{k\geq 0}$ be the sequence associated with the positive integer $n$. For a fixed $n$, define $(k_N)_{N\geq 1}$ by requiring that $s_{k_N}(n)$ is the $N$th prime term…

Let $(s_k(n))_{k\geq 0}$ be the sequence associated with the positive integer parameter $n$, and let $\mathcal{T}_p(j)$ denote the dilated Chebyshev polynomial of the first kind ev…

Let $H_a$ be the lexicographically first strictly increasing sequence beginning with $H_a(2)=a$ such that $H_a(n)$ is prime if and only if $n$ is prime. For primes $a>b\geq 3$, wri…