20 problems

Let $a$ and $b_{1}$ be relatively prime positive integers and put $b=-b_{1}^{2}$. Define the binary recurrence sequence by $u_{0}(a,b)=0$, $u_{1}(a,b)=1$, and…

For a positive integer $k$, let $S_k=\{n^k:n\in\mathbb{Z}_{>0}\}$; a set is Sidon if there are no nontrivial equal sums of two of its elements. Lander–Parkin–Selfridge's special-ca…

Let $p_r(n)$ denote the $r$-colored partition function, and define $L_{p_r}(d)$ and $M_{p_r,k}(d)$ analogously to the corresponding quantities for the partition and overpartition f…

Let $\overline{p}(n)$ denote the overpartition function, and define $M_{\overline{p},k}(d)$ analogously to $M_k(d)$ as the largest index whose overpartition number lies within $d$…

Let $pl(n)$ denote the plane partition function, counting plane partitions of $n$. A perfect power is an integer of the form $a^b$ with integers $a,b>1$. The plane-partition perfec…

For each fixed integer $d\geq 0$, define … and let $M_k(d)$ be the largest $n$ for which $p(n)$ is within $d$ of a $k$th power. Merca–Ono–Tsai's stabilization conjecture. There is…

The asymptotic conjecture for $M_k(d)$. For $k\geq 2$,

Merca–Ono–Tsai's finiteness conjecture. For fixed integers $k>1$ and $d\geq 0$, there are at most finitely many $n$ for which $\Delta_k(n)\leq d$. This finiteness assertion is used…

Sun's perfect-power repulsion conjecture. The partition function $p(n)$ repels perfect powers in the following senses: (i) for every $n\ge2$ and $k\ge2$, $p(n)\neq m^k$ for all int…

Partition-theoretic stabilization conjecture. For each non-negative integer $d$, there is a positive integer $N_d$ such that for $k\geq N_d$,

Refinement of the perfect-power repulsion conjecture. If $k>1$, then $M_k(0)=1$, and for every $\varepsilon>0$,

Schäffer's conjecture. If $(k,n)\notin S$, then the equation

Let $k\geq 1$, $h\geq 2$, and $\varepsilon>0$. A $B_h[g]$ set $A\subseteq(\mathbb{Z}^{+})^k$ is a set of $k$-th powers with counting function $A(x)$. Generalized Sidon-set conjectu…

Large prime divisor conjecture. There exists a prime divisor $p$ of this integer such that $p>z$. The paper presents this as an observation suggested by computations and proposes i…

Multiple-root conjecture. The polynomial $g_T(x)$ has multiple roots if and only if one of the following holds: (1) $T=(n-4)$ for $n\geq4$, with…

Let $r=2d$ with $1\leq d\leq 50$, let $p\geq 5$ be prime, and consider the equation … Here $d,x,z$ are integers. Alternating-cube perfect-power conjecture. The possible integer sol…

Nontrivially squared prime-pair conjecture. There are infinitely many consecutive primes $p$ and $q$ ($p>q$) which are nontrivially squared.

Trivially squared prime-pair conjecture. There are infinitely many consecutive primes $p$ and $q$ ($p>q$) which are trivially squared.

Let $F$ be a separable homogeneous cubic binary form with integer coefficients, let $d$ be a fixed integer with $d\geq 1$, and let $l$ be a prime number. Cubic binary-form perfect-…

Let $\varepsilon>0$, and let $a,b,x,y$ be positive integers with $x,y\ge 2$ and $a^x\ne b^y$. Quantitative Pillai conjecture. There exists a constant $\kappa(\varepsilon)>0$ such t…