15 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…

For the first Euclid–Mullin sequence, define $p_n$ to be the smallest prime dividing $p_1\cdots p_{n-1}+1$, with $p_1=2$. Shanks's conjecture. The sequence contains every prime. Eq…

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…

Let $a_1,\ldots,a_k$ be functions from a Hardy field with polynomial growth, and consider averages along the primes whose iterates are given by $\lfloor a_i(p)\rfloor$. Frantzikina…

Let $c$ be a positive non-integer, and let $p_n$ denote the $n$-th prime. Frantzikinakis's recurrence and convergence conjecture. The sequence $\lfloor p_n^c\rfloor$ should be good…

The sequences under consideration are integer parts of fractional powers, such as $\lfloor n^{3/2}\rfloor$ and $\lfloor n^{\sqrt{2}}\rfloor$, together with their linear combination…

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…

Let $p_1,p_2,\ldots$ be Chua's sequence, and let $n=p_1\cdots p_k$ be the product of its first $k$ terms. The next term $p_{k+1}$ is characterized by the existence of a divisor…

Let $(P_n)_{n\geq 1}$ denote Mullin's second sequence of primes. A prime is omitted if it does not occur among the terms $P_n$. Cox and van der Poorten's conjecture. Infinitely man…

Hardy–Wright's conjecture. The number of Fermat primes is finite.