47 problems

Catalan's conjecture. The only such solutions occur for $m=2$ and $n=3$, with

Rudin's conjecture. The maximal number of squares satisfies

Schäffer's conjecture. If $(k,n)\notin E$, then the only nontrivial integer solution of

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 $(F_n)_{n\geq 0}$ be the Fibonacci sequence, defined by $F_0=0$, $F_1=1$, and … A perfect power is a positive integer of the form $y^a$ with integers $y>1$ and $a>1$. Luca–Pate…

For $w>0$, consider the perfect powers lying in the interval $[w,w+\sqrt{w}]$. Loxton's conjecture. The number of perfect powers in this interval is bounded by an absolute constant…

For fixed $k\geq 2$, let $\mathcal{Q}_k(N;q,a)$ be the number of $k$-th powers among the terms $a+q,a+2q,\ldots,a+Nq$ of an arithmetic progression, and define … Here $\lessapprox$…

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…

For an integer $m\geq 2$, let $S_m$ be the set of $m$-th powers, and let a Hilbert cube be a set of the form $H(a_0;a_1,\ldots,a_d)$. Folklore conjecture. For each $m\geq 2$, the d…

Let $p_n$ denote the $n$th prime. The preceding problem asks whether there is an absolute constant $C$ such that infinitely many integers $n$ satisfy … and $p_{n+1}-p_n$ is a squar…

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 growth conjecture. For each integer $k>1$ and every real $\varepsilon>0$, one has

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…

Fibonacci perfect-power conjecture. The only Fibonacci numbers that are perfect powers are $0$, $1$, $8$, and $144$. This conjecture was confirmed by Bugeaud, Mignotte and Siksek u…

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 $\varepsilon>0$,

Perfect-power repulsion conjecture. If $k>1$ and $d\geq 0$, then there are at most finitely many $n$ for which

Euler's infinitude conjecture. The equation

Let $n otin{0,1}$ be a natural number, and let $A352991(n)$ denote the $n$th term of the sequence $A352991$. The conjecture. If $A352991(n)$ is a perfect power of an integer, t…

Let $A001292$ be the OEIS sequence considered in Kashihara's open problem number 30, and call an element a perfect power if it is an integer power with exponent greater than one. K…

Let $A352991$ be the OEIS sequence referenced in the source, and let a term of this sequence be a perfect power of an integer. Ripà's conjecture. All perfect powers belonging to…