Open-problem statements, with their sources attached.

It is likely that there are infinitely many primes $p$ such that $8p^2 - 1$ is also prime.

Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then

is not a perfect power?

Let $k_1 \geq k_2 \geq 3$. Are there only finitely many $n_2\geq n_1 + k_1$ such that

have the same prime factors?

Erdős thought perhaps if the two products have the same factors then $n_2 > 2(n_1 + k_1)$. It is an open question whether this is true when allowing a finite number of counterexamples.

Erdős was unable to prove that if the two products have the same factors then there must exist a prime between $n_1$ and $n_2$.

Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r < n < p_{r+1}$ all of whose prime factors are $< p_{r + 1} - p_r$.

If $n(n+1)=2^k3^lm$, where $(m,6)=1$, then is it true that $\limsup_{n\to \infty} \frac{2^k3^l}{n\log n}=\infty$?

Is $2^n + 1$ powerful for finitely many $n$?

Is $2^n - 1$ powerful for finitely many $n$?

Is $n! + 1$ powerful for finitely many $n$?

Is $n! - 1$ powerful for finitely many $n$?

Let $A=\{n_1 < n_2 < \cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$). Are there only finitely many three-term progressions of consecutive terms $n_k,n_{k+1},n_{k+2}$?

If $r≥4$ then can the sum of $r-2$ coprime $r$-powerful numbers ever be itself $r$-powerful?

If $r≥4$ are there infinitely many sums of $r-2$ coprime $r$-powerful numbers that are themselves $r$-powerful?

Are there infinitely many triples of coprime $3$-powerful numbers $a, b, c$ such that $a + b = c$?

Let $r \ge 3$. Is it true that the set of integers which are the sum of at most $r$ $r$-powerful numbers has density $0$?

Is it true that the set of integers which are the sum of at most three cubes has density $0$?

It is not known if all large integers are the sum of at most $r$-many $r$-powerful numbers.

Is there some constant $c > 0$ such that $h(n) < (\log n)^{c + o(1)}$ and, for infinitely many $n$, $h(n) > (\log n)^{c - o(1)}$.

Let $A$ be the set of powerful numbers. Is is true that $1_A\ast 1_A(n)=n^{o(1)}$ for every $n$?