33 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 $\mathcal{B}_{2m}$ be the elliptic Bernoulli numbers, and let $w$, $\eta$, $g_2$, and $g_3$ denote the associated Weierstrass quantities. Positivity conjecture. The elliptic Be…

Bernoulli–quadratic-form conjecture. For every prime $p$,

For an even integer $2m$, let $P(2m)$ be the set of primes $p$ such that $p-1\mid 2m$, so that … where $D_{2m}$ is the denominator of the Bernoulli number $B_{2m}$. Let…

Let $b_m(n)$ denote the Bernoulli partition entries described in the preceding discussion, and let $B_{2m}$ be the Bernoulli numbers. For each column index $n\geq 2$, define the as…

Let $p\ge 5$ be a prime, and let $\\{u_n\\}_{n\ge 0}$ be one of the sequences $\mathbf{B}$ and $\mathbf{F}$, defined by … or … Here $\left(\frac{p}{3}\right)$ is the Legendre symbo…

Bernoulli--Seki nonvanishing consequence. Let $k\geq3$ be an odd integer. Then, there exist infinitely many primes $p$ greater than $k$ such that $p$ does not divide $B_{p-k}$. The…

A prime $p$ is regular if it does not divide any of the Bernoulli--Seki numbers $B_2,B_4,\ldots,B_{p-3}$. Infinitude conjecture for regular primes. There exist infinitely many regu…

Let $p$ be an odd prime and let … be the Artin–Hasse exponential series in $\mathbb F_p[[X]]$. For an integer $n$, let $B_n$ denote the $n$-th Bernoulli number. The conjectural rec…

Bernoulli-number conjecture. The coefficients satisfy

Let $k$ and $\ell$ be fixed natural numbers, and define … Here $B_m$ denotes the $m$th Bernoulli number; the variable $Z$ replaces $z^{2\ell}$ in the one-variable generalization of…

Conjecture on the zeros of $R_{2k+1,p}$. The only real root of $R_{2k+1,p}(z)$ is $z=0$, with multiplicity $2$; moreover, its non-real roots are simple and lie on the circle…

Let $\mathcal P(g,t)$ and $\mathcal Q_{\alpha,\beta}$ be the prime sets defined earlier in the paper, and let $\delta(g,t)$ and $\rho_{\alpha,\beta}(a,d)$ be the corresponding cons…

Let $R_B(x)$ denote the number of $B$-regular primes up to $x$, where an odd prime is $B$-regular if it divides none of the numerators of $B_2,B_4,\ldots,B_{p-3}$. Siegel's conject…

Let $\chi$ be a Dirichlet character, and let $\operatorname{ord}(\chi)$ denote its order. A prime $p$ is $\chi$-regular when the relevant even twists $\chi\omega^i$ have vanishing…

Bernoulli congruences for $R_8^{(m)}(p)$, $R_9^{(m)}(p)$, and $R_{10}^{(m)}(p)$.

Let $p>3$ be a prime, let $n$ be a positive even integer, and write $p^l\mathrel{\|}n$ to mean that $p^l\mid n$ but $p^{l+1}\nmid n$. Assume that $p-1\nmid n$. Write…

Let $n$ be a positive integer and let $B_{n-1}$ denote the $(n-1)$-st Bernoulli number. Agoh–Giuga conjecture. The integer $n$ is prime if and only if … This conjecture characteriz…

Let $\mathbb{N}$ denote the positive integers. For each $m,n\in\mathbb{N}$, let $R_n^{(m,1)}$ and $S_n^{(m,1)}$ be the finite multiple harmonic sum quantities defined in the source…

Terminal-coefficient conjecture. The polynomials and constants are given by

Chen's conjecture. For every $t>0$ and $m\in\mathbb{N}_0$, one has

For every odd integer $k\geq 3$, define the element $(B_{p-k}\pmod p)_p\in\mathcal{A}_{\mathbb{Q}}$, where $B_m$ denotes the $m$-th Bernoulli number. Bernoulli finite zeta nonvanis…

Non-vanishing conjecture. There exist infinitely many primes $p$ such that