32 problems

Sun's sequence of polynomials is defined by … A polynomial sequence $\{f_n(q)\}_{n\geq 0}$ is q-log-convex if, for every $n\geq 1$, the polynomial $f_{n+1}(q)f_{n-1}(q)-f_n^2(q)$ h…

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 $N\rightarrow\infty$.

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 $f\in\mathbb{Z}[x]$ be a polynomial of degree greater than one, and let $A_f=(f(1),f(2),f(3),\ldots)$ be its associated integer sequence. Write $g_f=g_{A_f}$, where $g_A$ is th…

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…

Chapoton–Han's determinant conjecture. The determinant satisfies

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

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…

Let $cmu$ denote the Möbius function, and let $f e c1g^2$ be a polynomial in $cZ[x]$, where $c1 e 0$ belongs to $cQ$ and $g e 0$ belongs to $cZ[x]$. Chowla's conjecture. One has ……

Let $f\in\mathbb{Z}[X]$, let … and let … A prime is said to divide $Q_f(N)$ exactly once when its exponent in the prime factorization of $Q_f(N)$ is one. Sah's conjecture. The size…

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

The cubes' $\Lambda_4$ conjecture. The set of cubes

Cilleruelo–Córdoba conjecture. There is a constant $c(\gamma)>0$, depending only on $\gamma$, such that

Let $lambda:\mathbb{N}\to\{-1,1\}$ be the Liouville function, defined by $\lambda(n)=(-1)^{\Omega(n)}$, where $\Omega(n)$ counts prime factors with multiplicity. Let…

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…

Let $\ell\in\mathbb{N}$ and let $p_1,\ldots,p_\ell\in\mathbb{R}[x]$. For a family of real polynomials, $\mathbb{R}\backslash\mathbb{Q}_*$-independence means $V$-independence with…

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…

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…

Optimality conjecture. This upper bound is best possible apart from a logarithmic factor. The conjecture asserts near-optimality of the square-root discrepancy bound for almost all…

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