145 problems

Strengthened truncated-mean inequality. For every $-1<t<1$,

Let $\beta(L)$ denote the beta-value associated with a list $L$, and let list concatenation be written by juxtaposition. Let $m,n$ be non-negative integers with $n>m$, and let $L$…

Let $(m_1,n_1)$ and $(m_2,n_2)$ be pairs of non-negative integers, where $(m_1,n_1)$ is strictly better balanced than $(m_2,n_2)$, meaning that the first pair is strictly closer to…

Nonnegative-coefficient conjecture. If $r\in\mathbb N$ is arbitrary but fixed, then $h(t,a;r)$, viewed as a polynomial in $(t,a)$, has nonnegative coefficients. This would consider…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, form the three q…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, use…

Let $X$, $Y$, and $Z$ be upward closed subsets of $Q_n$. It is conjectured that … Notation: the poset $Q_n$ is the power set of a set with $n$ elements, ordered by subset relation.…

For $0\leq\mu_1\leq\cdots\leq\mu_n$ and $\lambda>\sum_{i=1}^n\mu_i$, let $X_1,\ldots,X_n$ range over independent nonnegative random variables satisfying $\mathbb{E}X_i=\mu_i$. Defi…

For every integer $n\ge 1$, every $-1<\alpha<\beta$, and every $0<c\le 1$, let $\pi_n$ be the monic degree-$n$ polynomial orthogonal on $[-c,c]$ with respect to the Jacobi weight…

Half-Eulerian inequality-generation conjecture. Every linear form that is nonnegative on the flag vectors of all half-Eulerian posets is the sum of a linear form that is nonnegativ…

Let $\beta(L)$ denote the beta-value associated with a list $L$. A list is balanced if its entries are either $k$ or $k+1$ for some non-negative integer $k$. Let $L$, $M$, and…

Let $(a(n))_{n\geq 1}$ be the EKG sequence defined by $a(1)=1$, $a(2)=2$, and, for $n\geq 3$, choosing the smallest unused natural number sharing a nontrivial common divisor with…

Sufficient inequalities for Stolarsky's bounds. The inequalities

Stolarsky's bounds conjecture. For all $(a,b,c)\in R_{+}^{3}$,

Let $n\geq 3$ and consider extremizers of the $p$-Stam inequality among normalized monic real-rooted polynomial pairs $(f,g)$ of degree $n$; write $d_{PQ}$ for their pair mismatch…

For each $n\geq 3$, consider the $p$-Stam inequality for monic real-rooted pairs of degree $n$, with critical exponent denoted by $p^*$. $p$-Stam phase transition conjecture. For a…

For $g\geq2$, $n\geq1$, and $d_1,\ldots,d_n\geq1$ satisfying $|\mathbf d|=3g-3+n$, let $C(\mathbf d)$ denote the primitive normalized Witten intersection number, and write…

B1nger--Seeger's 200 Euro Conjecture. There exists $x\in\mathbb{R}^n\setminus\{0\}$ such that

Rump's 100 Euro Conjecture. There exists $x\in\mathbb{R}^n\setminus\{0\}$ such that

Let $\widehat{m}_{1/2}(x,\alpha)$ denote the logarithmic error term for the shifted interpolation parameter $\alpha$, let $\widehat{D}_{1/2}$ be its stated domain, and let…

Subgroup-counting positivity conjecture. If $\ell=3$ or $\ell=4$, then for every $n\geq a_{\ell}$ and every real $x\geq 1$,

D'Arcais positivity conjecture. For every integer $n\geq 6$ and every real $x\geq 1$,

Monotonicity conjecture. For any $n\in\mathbf{N}$, we have

Let $B_1(x)$ be the first Bernoulli polynomial, let $\psi$ denote the digamma function, and let $\gamma=-\psi(1)$ be the Euler–Mascheroni constant. Define … For $0<x<1$, the inequa…

Marques–Trojovský conjecture. The function $\Psi(x)$ satisfies: (i) $\Psi(x)=21$ for all $x\in[0.9,1.3]$; (ii) $\Psi(x)=149$ for all $x\in[0.5,0.8]$; (iii) $\Psi(x)=59{,}875$ for a…