15 problems

Let $\mathcal{A}$ be the set of positive integers $N$ for which the corresponding triple $N,N+d,N+2d$ is a three-term arithmetic progression of consecutive powerful numbers. Partit…

Erdős–Mollin–Walsh conjecture. No three consecutive integers are all powerful.

For an integer $m$, define its squarefree kernel, or radical, by … Let $a$, $b$, and $c$ be integers satisfying $a+b=c$ and $\operatorname{gcd}(a,b)=1$. The abc-conjecture. For eve…

Richert's conjecture. The most optimistic conjecture is that $c=1/6+\varepsilon$ holds.

Let $\mathcal{A}$ be the set of positive integers $N$ for which some $d\in\mathbb{N}$ makes $N,N+d,N+2d$ a three-term arithmetic progression of consecutive powerful numbers. Let…

A positive integer $n$ is powerful if every prime divisor $p$ of $n$ satisfies $p^2\mid n$. A three-term arithmetic progression of consecutive powerful numbers is a triple…

Let $\mathcal{VB}^1$ denote the set of powerful numbers, and write $\mathcal{VB}^1+1=\{n+1:n\in\mathcal{VB}^1\}$. Erdős's consecutive-powerful-numbers conjecture. … The conjecture…

Let ${\mathcal B}$ be the set of natural numbers contained in at least one bad interval, let $\mathcal{VB}$ be the set of natural numbers contained in at least one very bad interva…

Non-powerfulness conjecture. For every integer $x>1$ and every integer $n>2$, the number

Schinzel–Tijdeman conjecture. There are at most finitely many integers $w$ such that $P(w)$ is a powerful number.

A positive integer is powerful if every prime divisor $p$ satisfies $p^2\mid n$. For an integer $N$, the three consecutive integers are $N$, $N+1$, and $N+2$. Erdős's conjecture. T…

Let $k\ge 2$ be an integer, let $x\ge 1$ be real, and let $Q_k(t)$ denote the number of $k$-full numbers at most $t$. A number is $k$-full if every exponent in its prime factorizat…

A positive integer is powerful if every prime divisor of it occurs with exponent at least two. Powerful-number prime-gap conjecture. If $a>1$ is powerful, then there is a prime $p$…

Golomb–Erdős conjecture. There are only finitely many positive integers $n$ such that $n$, $n+1$, and $n+2$ are all powerful.

Let $P$ be a polynomial with rational coefficients. Schinzel–Tijdeman conjecture. If $P$ has at least three simple zeros, then the equation … has only finitely many solutions in in…