56 problems

For an integer $p>1$, let $\mathbf{B}_{p-1}$ denote the $(p-1)$-st Bernoulli number. Agoh–Giuga conjecture. The integer $p$ is an odd prime if and only if … This is a Bernoulli-num…

Agoh's conjecture. The integer $n$ is prime if and only if

Let $B_n$ be the $n$-th Bernoulli number. Sun's conjecture. The sequence … is strictly increasing. The paper proves this, confirming Sun's conjecture; Luca and Stănică independentl…

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…

For positive integers $k$ and $\ell$, define the polynomial … where $B_m$ denotes the $m$th Bernoulli number. Maji–Sarkar conjecture. All the non-real zeros of $R_{k,\ell}(x)$ lie…

Let $r\in\mathbb{Z}^{+}$ and let $p>3$ be prime. Write $B_{p-3}$ for the Bernoulli number indexed by $p-3$. Sun's conjecture. … This extends the corresponding congruence modulo $p$…

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…

Gessel–Lengyel equality conjecture. Equality should always hold in both of these inequalities. The claim was conjectured by I. M. Gessel and T. Lengyel in 2001, but the present sou…

For a positive integer $p$, let $B_r$ denote the Bernoulli numbers, let $H_n=\sum_{k=1}^n 1/k$ be the harmonic numbers, and let $H_n^{(2)}=\sum_{k=1}^n 1/k^2$ be the second-order h…

For an even integer $n$, let $B_n$ denote the $n$th Bernoulli number, let $|\cdot|_p$ be the $p$-adic absolute value, let $\Psi^{\rm irr}_1$ be the set of irregular pairs, and for…

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…

For a prime $p$, consider the Bernoulli numbers and, equivalently, the values of the Riemann zeta function at odd negative integers, reduced modulo $p$. Even distribution conjectur…

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 $\mathcal{B}_{2m}$ 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 $q$ be a prime. It is Bernoulli regular if it does not divide any of $t_1,\dots,t_{(q-3)/2}$, where $t_n$ is defined by … with $\gcd(t_n,b_n)=1$. Infinitude of Bernoulli regula…

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…

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…