9 problems

Bacher's conjecture. Then $A(p)=(l-1)p+1$.

$\Delta$-conjecture. The following properties are equivalent:

Let $M$ be a fixed positive integer and let $c$ be an integer with $0\le c<M$. An irregular pair is a pair $(p,t)$ such that $p$ is prime, $t$ is even, and $p$ divides the Bernoull…

Let $\ell$ be a prime and let $1\leq a<d$ with $(a,d)=1$. A prime is $\ell$-Genocchi regular if it is not $\ell$-Genocchi irregular. Positive-density conjecture. If $\ell$ is odd,…

Let $\ell$ be an odd prime, let $1\leq a<d$ with $(a,d)=1$, and let $\mathcal P_G(d,a)(x)$ count the primes $p\leq x$ satisfying $p\equiv a\pmod d$ that are $\ell$-Genocchi irregul…

Let $a$ and $d$ be coprime positive integers, and consider primes in the primitive residue class $a\pmod d$. The G-regular-prime density conjecture. The subset of G-regular primes…

Given coprime positive integers $a$ and $d$, let ${\mathcal P}_G(d,a)$ be the set of G-irregular primes congruent to $a$ modulo $d$, and let $\pi(x;d,a)$ count primes $p\le x$ with…

Let ${\mathcal P}_G$ be the set of G-irregular primes, and write ${\mathcal P}_G(x)$ for the number of its elements not exceeding $x$. Let $A$ be the Artin constant, … and let…

Let $S_m(n)=\sum_{j=1}^{n-1}j^m$, and let $B_m$ denote the $m$th Bernoulli number. An irregular prime is a prime dividing the numerator of some Bernoulli number $B_{2r}$ with…