190 problems

Let $\mathrm{den}(n,x)$ be the denominator polynomial associated with sums of reciprocals of subsum polynomials over partitions of $n$. Log-concavity conjecture. The sequence of co…

Let $P_n(z)$ be the sorted binomial polynomial with parameter $n$. Irreducibility conjecture. $P_n(z)$ is irreducible over $\mathbb{Q}$ for all even $n$. The analogous factorizatio…

Alternating-sign conjecture. Like the Bernoulli numbers, the signs of the even-indexed Bernoulli-Catalan numbers $\tau_{2n}^*$ alternate.

Bouniakowsky's conjecture. There are infinitely many primes of the form $P(z)$ for $z\in\mathbb{Z}^+$. This conjecture predicts that every integer polynomial satisfying the stated…

For $m,n\in\mathbb{N}$, let $A_n^{(m)}$ be the $n\times n$ matrix defined by … Here $\delta_{ij}$ is the Kronecker delta. Sun's determinant-form conjecture. Sun conjectured that ……

Let $p$ be an odd prime, let $\left(\frac{\cdot}{p}\right)$ denote the Legendre symbol modulo $p$, and define the Fekete polynomial … Montgomery's bounds concern the maximum of…

Let $M_1=M(\{0,\pm1\})$ be the closure in $\mathbf{C}$ of the set of zeros of polynomials with coefficients in $\{0,\pm1\}$ and nonzero constant term. Write…

Schur's finiteness conjecture. There are only finitely many pairs $(a,p)$ with $a<0$ such that $P_{p,a}(X)$ has no linear factor and is not irreducible.

Let $m\in\mathbb{N}$ and let $f_1,\ldots,f_m\in\mathbb{Z}[x]$ be polynomials. Suppose that there is no prime number $p$ such that … for every $n\in\mathbb{N}$. Schinzel–Sierpiński…

For a polynomial $f\in\mathbb{Q}[X]$ and a prime $p$, let $\bar{f}_p\in\mathbb{F}_p[X]$ be its reduction modulo $p$. A polynomial over a field is dynamically irreducible if every i…

Let $n\geq 2$, let $Q_n$ be the polynomial studied in the paper, and write … For a polynomial with rational coefficients, let $\operatorname{Gal}(Q_n)$ denote its Galois group and…

Univalence conjecture. The polynomial $F$ is univalent in the unit disk $\mathbb D$. Numerical evidence supports the assertion for all $N$, while it is established in the cases…

Simplicity conjecture. All nonzero roots of $W_N^{[m,k,l]}(z)$ are simple.

Generalized Laguerre conjecture. Given $t>0$, the polynomial $h_n(x;t)$ satisfies

Baker's conjecture. The exponent in this bound should be improvable to $1+o(1)$, meaning that one should have

Distinct-zeros conjecture. The positive zeros of $\Sigma(R_1)$ and $\Sigma(R_2)$ are distinct.

Let $p_N$ be a polynomial of degree $N$ in the normalized univalent class $\mathcal S$, and define its Koebe radius by … The Koebe radius is the infimum of the boundary modulus amo…

Let $f\in\mathbb{Z}[T]$ be primitive and squarefree, let $\mathcal{A}=a+N\mathbb{Z}\subseteq\mathbb{Z}$ be an arithmetic progression, and let…

Bouniakowski–Schinzel conjecture. If $f(x)$ represents primes, then $f(x)$ has infinitely many prime values.

Define the Gandhi polynomials by … with $\widetilde P_1(z)=1$, and let $G_{2r}$ denote the Genocchi numbers. Gandhi's conjecture. … The conjecture concerns a representation of the…

Let … The values $S_1(n)=n\binom{2n}{n}$ suggest the following claim. Odd absolute-binomial sum conjecture. For every relevant nonnegative integer $r$, $S_{2r+1}(n)$ equals…

Equivalence conjecture. All non-real solutions are equivalent to real solutions.

Let $R(x)=1+\sum_{i\geq 2}R_ix^i$, and define $C_m$ by $C_0=1$, $C_1=0$, and … For $n\geq 0$, let $\Sigma_{k,2n}$ be the sum of the terms of weight $k+1-2n$ in $\Sigma_k$. A polyno…

For $n\geq 0$, let $\Sigma_{k,2n}$ denote the component of weight $k+1-2n$ in the Kerov character polynomial $\Sigma_k$, and write $[R_2^i]\Sigma_{k,2n}$ for the coefficient of…

Uniqueness conjecture for extremal polynomials. The extremal polynomial in Smale's mean value conjecture is unique up to affine equivalence.