7 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…

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 $p^2\mid n$ for every prime $p$ dividing $n$. Three positive integers are consecutive powerful numbers if they have the form $n,n+1,n+2$ and e…

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 $p$…

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…