Open-problem statements, with their sources attached.

A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810

[Needed to index shift in order to avoid trivial case $n = 0$, where the conjecture is trivially false.]

Let $\epsilon > 0$. Is there some set $A\subset\mathbb{N}$ of density $> 1 - \epsilon$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$?

Is there some set $A\subset\{1, ..., N\}$ of size $\geq (1 - o(1))N$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$?

Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that $g(n) > n^{1-\epsilon}$?

Are there infinitely many $n$ such that, for all $k\geq 1$

Is it true that, for any $a \in \mathbb{Z}$, there are infinitely many $n$ such that $\phi(n) | n + a$?

When $n > 1$, Lehmer conjectured that $\phi(n) | n - 1$ if and only if $n$ is prime.

Erdős Problem 829 (open).* Let $A \subseteq \mathbb{N}$ be the set of perfect cubes. Is it true that $(1_A \ast 1_A)(n) \ll (\log n)^{O(1)}$? That is, does there exist a natural number $C$ such that the number of representations of $n$ as a sum of two cubes is $O((\log n)^C)$ as $n \to \infty$?

Erdos Problem 830, Part 1* We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. Are there infinitely many amicable pairs?

Erdos Problem 830, Part 2* We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$. If $A(x)$ counts the number of amicable $1\leq a\leq b\leq x$ then is it true that $A(x) > x^{1-o(1)}?$

Is it true that, for every integer $t\geq1$, there is some integer $a$ such that ${n \choose k} = a$ with $1\leq k \le \frac{n}{2}$ has exactly $t$ solutions?

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?

Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n = t$ has no solutions for $n \le x$.

Is it true that $r(x) \to \infty$?

Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n = t$ has no solutions for $n \le x$.

Is it true that $r(x) / \log x \to \infty$?

Erdős Problem 855 (Segal's conjecture): $\pi(x + y) \le \pi(x) + \pi(y)$ for sufficiently large $x, y$.

The density of the divisor sum set is asymptotically equivalent to $c_1 / \log(t)^{c_2}$.

Let $A = \{a_1 < a_2 < \dots\} \subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that $[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X$, where the left-hand side is the least common multiple. Is it true that, for every $\epsilon > 0$, there exists some $k$ such that $F(A,X,k) < X^\epsilon$?

Is it true that, for every $k \geq 1$, there exist integers $N_1 < \dots < N_k$ such that $|\cap_i D(N_i)| \geq k$?

Let $\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\epsilon})$ is $O_\epsilon(1)$?

Erdős attributes this conjecture to Ruzsa.

Is there an absolute constant $K$ such that, for every $C > 0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + C n^{\frac{1}{4}})$.