19 problems

Let $(p,l)$ and $(q,l')$ be irregular pairs, and suppose that $q$ is an irregular prime divisor of $l-1$. Here $p$, $q$ are odd primes and $l$, $l'$ are even irregular indices. Ext…

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 … -irregular when it is irregular in the sense determined by … -irregular primes should be distributed evenly. The paper presents this as an extension of conjectures about irre…

Let $B_{2m}$ denote the Bernoulli number of even index, and let $p$ be an odd prime. Siegel's conjecture. The numerators of the Bernoulli numbers $B_{2m}$ should be evenly distribu…

Let $p$ be a prime. The Bernoulli numbers, equivalently the values of the Riemann zeta function at odd negative integers, are considered modulo $p$. Even-distribution conjecture. T…

Let $\mathcal{P}$ be the set of all primes, and call a prime irregular if it divides one of $B_2,B_4,\ldots,B_{p-3}$. Irregular-prime density conjecture. The relative density of th…

For a prime $p\geq5$, let $\eth_p$ be the least odd integer $k\geq3$ for which $p$ does not divide $B_{p-k}$, and let $i(p)$ be the index of irregularity. Boundedness conjecture fo…

Let $A$ be the Artin constant, let $\ell$ be a prime, and let $\mathcal P_G(x)$ count the $\ell$-Genocchi irregular primes up to $x$. Explicit ell-Genocchi asymptotic conjecture. A…

Let $\ell$ be a prime, let $\mathcal P_G(x)$ count the $\ell$-Genocchi irregular primes up to $x$, and let $\delta(\ell^2,2)$ be the explicit constant defined in the paper. ell-Gen…

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…

Fix a prime $\ell$, an integer $\varepsilon\in\{-1,1\}$, and let $\mathcal P_{H^{\varepsilon}}(x)$ count the $H^{\varepsilon}$-irregular primes up to $x$. H^{varepsilon}-irregular-…

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…

For positive integers $a,d$ with $\gcd(a,d)=1$, let ${\mathcal P}_E(d,a)$ be the set of E-irregular primes congruent to $a$ modulo $d$. Let ${\mathcal P}_E(d,a)(x)$ count its eleme…

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…