10 problems

Bernoulli-moment sign conjectures. For every $k\in{\mathbb N}$, both the weak form

For $k\geq1$, define … and let $\mathcal{S}_k$ be the corresponding computable subsets, with … … Second- and third-derivative completeness conjecture. One has … The paper proves fi…

Define … Radical characterization conjecture. One has … The surrounding results show that $\overline{\mathcal{R}}$ is finite and constrain its elements; the conjecture remains unre…

The Bernoulli polynomials $\mathbf{B}_n(x)$ are defined by … and let … Finiteness and completeness conjecture. The set $\overline{\mathcal{S}}$ is finite and … This conjecture asse…

For each positive integer $n$, let $b_n(u)$ denote the factorially normalized Bernoulli polynomial of degree $n$, and consider the polynomial probability density $1-2^n b_n(u)$ on…

Let $b_n$ be the factorially normalized Bernoulli polynomial of degree $n$, and let $\delta_{k:2n}$ denote the difference between the uniform distribution on $\{1,\dots,2n\}$ and t…

Let $r\geq 1$, and let $\bar F_{r,2}(x_1,x_2)$ be the determinant polynomial defined in the paper for $D=2$. Factorization conjecture. One has … The formula was supported by comput…

Let $z$ and $N$ be integers with $\gcd(z,N)=1$. Let $B_1$ and $B_2$ denote the first and second Bernoulli polynomials, respectively, and interpret $z(k-\ell)\bmod N$ as the residue…

For a positive integer $n$, define … where $s_p(n)$ is the sum of the base-$p$ digits of $n$. Let $\omega(m)$ denote the number of distinct prime factors of $m$. Kellner's conjectu…

For a positive integer $n$, let $s_p(n)$ denote the sum of the base-$p$ digits of $n$, and define … Let $P(m)$ be the largest prime factor of $m$. Kellner's conjecture. For $n>192$…