26 problems

For each $n$, write … where $P_n(q)$ is the normalized polynomial associated with colored interlacing triangles. Polynomiality conjecture. For each fixed $k\geq 0$, there is an…

For each fixed $k\geq 0$, let $a_k(n)$ be the coefficient of $q^k$ in $P_n(q)$, regarded as a polynomial in $n$ as in the polynomiality conjecture. Integrality and leading-coeffici…

Let $N\geq 1$ be a natural number, and let $P_{2n+2l}$ be a sequence of integer polynomials defined in the source's equation (with coefficients bounded in modulus by $N$). Its limi…

Let $W=N/\Lambda$ be a connected nilmanifold, let $Y=\pi(H)$ be a connected subnilmanifold of $W$, where $H$ is a connected closed subgroup of $N$ and $\pi:N\to W$ is the quotient…

Quantitative Erdős–Kac conjecture for irreducible quadratics. The Kolmogorov distance between $X_N$ and the standard Gaussian $\mathcal{N}(0,1)$ is…

Erdős–Kac conjecture for integer parts of polynomials. The random variable $X_N$ converges in distribution to the standard Gaussian $\mathcal{N}(0,1)$ as $N\rightarrow\infty$.

Limiting alternative-measure conjecture. The sequence has a limit, and

Let $R$ be obtained from the set $\{x^k+1:x\in\mathbb{N}\}$ by changing $o(X^{1/k})$ elements up to $X$, where $k\geq 2$. A set $S\subset\mathbb{N}$ is multiplicatively irreducible…

Sah's conjecture. As $N\to\infty$,

Let $\{P_m(z)\}_{m\geq 0}$ be a sequence of functions of $z$ generated by … Here $a,b,c\in\mathbb{R}\backslash\{0\}$ and $b^2-4ac\leq 0$. Let $t_1$ be the zero of $at^2+bt+c$ havin…

The conjecture for $\psi_F$. One should have

Let $s_Z(n)$ denote the sum of digits of $n$ in Zeckendorf base, let $\mathcal{S}_Z=(s_Z(n)\bmod 2)_n$, and let $\mathcal{S}_{Z,P}$ be the sequence obtained by restricting…

Let $\mathcal{T}_P$ denote the Thue–Morse sequence along a polynomial subsequence, where $P$ is a polynomial of degree $d$. Write $M(\mathcal{T}_P,N)$ for its maximum order complex…

Cassaigne et al.'s conjecture. The values $\lambda(f(n))$ change sign infinitely often.

Let $f\in\mathbb Z[x]$ be irreducible over $\mathbb Q$ of degree $d\ge 2$, and define … where $\operatorname{rad}(n)$ is the product of the distinct primes dividing $n$. Radical LC…

Let $f\in\mathbb Z[x]$ be irreducible over $\mathbb Q$ of degree $d\ge 2$, and define … ignoring zero values and taking the least common multiple of an empty set to be $1$. Cilleru…

Cilleruelo's conjecture. As $N\to\infty$,

Let $\{f_i(q)\}_{i\geq 0}$ be any of the polynomial sequences defined in Example $\mathrm{basic\text{-}qSM}$. A sequence is infinitely $q$-log-convex if every iterate under … has c…

Let $P(x)\in\mathbb{Z}[x]$ be square-free. The square-free sieve conjecture. The set of integers $n$ for which $P(n)$ is divisible by the square of a prime larger than $\sqrt n$ ha…

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

Write … Thus $p_{k,j}(t)$ denotes the coefficient functions in this expansion. Coefficient-shape conjecture. The coefficients satisfy … where … with rational positive nonzero numbe…

Polynomial-factorization conjecture. There exist polynomials $P_k(t,x)$ in $x$ of degree $k$, with coefficients rationally depending on $t$, such that

Real-rootedness conjecture for $Q_n$. For any $n\geq 1$, the polynomial $Q_n(x)$ has only real zeros. This conjecture is motivated by Brändén's theorem that the operator sending co…

Let $k$ be a field, let $\{\mathbf e_i\}_{1\leq i\leq \ell}$ be the standard basis of $k^\ell$, and let $\mathbf B=(B_1,\ldots,B_\ell)$ be a delta $\ell$-tuple with…

Let $p$ be the characteristic of the finite field ${\mathbb F_q}$, let ${\mathbb K}_\infty$ be the associated completion, and let ${\mathbb T}$ be the relevant torus. Let … be a po…