5 solved out of 12 shown (show only solved or open)
SOLVED - $1000 Can the smallest modulus of a covering system be arbitrarily large? Described by Erdős as 'perhaps my favourite problem'. Hough [Ho15], building on work of Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], has shown the answer is no: the smallest modulus must be at most$10^{18}$. An alternative, simpler, proof was given by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22], who improved the bound on the smallest modulus to$616000$. SOLVED -$10000
For any $C>0$ are there infinitely many $n$ such that $p_{n+1}-p_n> C\frac{\log\log n\log\log\log\log n}{(\log\log \log n)^2}\log n?$
The peculiar quantitative form of Erdős' question was motivated by an old result of Rankin [Ra38], who proved there exists some constant $C>0$ such that the claim holds. Solved by Maynard [Ma16] and Ford, Green, Konyagin, and Tao [FGKT16]. The best bound available, due to all five authors [FGKMT18], is that there are infinitely many $n$ such that $p_{n+1}-p_n\gg \frac{\log\log n\log\log\log\log n}{\log\log \log n}\log n.$ The likely truth is a lower bound like $\gg(\log n)^2$. In [Er97c] Erdős revised the value of this problem to \$5000 and reserved the \$10000 for a lower bound of $>(\log n)^{1+c}$ for some $c>0$.

OPEN
Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that $\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?$
Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\log n$. This problem asks whether $S=[0,\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:
• $\infty\in S$ by Westzynthius' result [We31] on large prime gaps,
• $0\in S$ by the work of Goldston, Pintz, and Yildirim [GPY09] on small prime gaps,
• Erdős [Er55] and Ricci [Ri56] independently showed that $S$ has positive Lebesgue measure,
• Hildebrand and Maier [HiMa88] showed that $S$ contains arbitrarily large (finite) numbers,
• Pintz [Pi16] showed that there exists some small constant $c>0$ such that $[0,c]\subset S$,
• Banks, Freiberg, and Maynard [BFM16] showed that at least $12.5\%$ of $[0,\infty)$ belongs to $S$,
• Merikoski [Me20] showed that at least $1/3$ of $[0,\infty)$ belongs to $S$, and that $S$ has bounded gaps.
In [Er97c] Erdős asks whether $S$ is everywhere dense.
SOLVED - $100 Let$d_n=p_{n+1}-p_n$. Are there infinitely many$n$such that$d_n<d_{n+1}<d_{n+2}$? Conjectured by Erdős and Turán [ErTu48]. Shockingly Erdős offered \$25000 for a disproof of this, but as he comments, it 'is certainly true'.

Indeed, the answer is yes, as proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]. They in fact prove that, for any $m\geq 1$, there are infinitely many $n$ such that $d_n<d_{n+1}<\cdots <d_{n+m}$ and infinitely many $n$ such that $d_n> d_{n+1}>\cdots >d_{n+m}.$

Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.
SOLVED - $250 Let$n\geq 1$and $A=\{a_1<\cdots <a_{\phi(n)}\}=\{ 1\leq m<n : (m,n)=1\}.$ Is it true that $\sum_{1\leq k<\phi(n)}(a_{k+1}-a_k)^2 \ll \frac{n^2}{\phi(n)}?$ The answer is yes, as proved by Montgomery and Vaughan [MoVa86], who in fact found the correct order of magnitude with the power$2$replaced by any$\gamma\geq 1$(which was also asked by Erdős in [Er73]). OPEN Let$d_n=p_{n+1}-p_n$. Prove that $\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.$ Cramer proved an upper bound of$O(N(\log N)^4)$conditional on the Riemann hypothesis. The prime number theorem immediately implies a lower bound of$\gg N(\log N)^2$. OPEN For every$c\geq 0$the density$f(c)$of integers for which $\frac{p_{n+1}-p_n}{\log n}< c$ exists and is a continuous function of$c$. OPEN Let$N_k=2\cdot 3\cdots p_k$and$\{a_1<a_2<\cdots <a_{\phi(N_k)}\}$be the integers$<N_k$which are relatively prime to$N_k$. Then, for any$c\geq 0$, the limit $\frac{\#\{ a_i-a_{i-1}\leq c \frac{N_k}{\phi(N_k)} : 2\leq i\leq \phi(N_k)\}}{\phi(N_k)}$ exists and is a continuous function of$c$. OPEN Let$f(n)$count the number of solutions to$n=p+2^k$for prime$p$and$k\geq 0$. Is it true that$f(n)=o(\log n)$? Erdős [Er50] proved that there are infinitely many$n$such that$f(n)\gg \log\log n$. Erdős could not even prove that there do not exist infinitely many integers$n$such that for all$2^k<n$the number$n-2^k$is prime (probably$n=105$is the largest such integer). See also [237]. SOLVED Let$A\subseteq \mathbb{N}$be a set such that$\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N$for all large$N$. Let$f(n)$count the number of solutions to$n=p+a$for$p$prime and$a\in A$. Is it true that$\limsup f(n)=\infty$? Erdős [Er50] proved this when$A=\{2^k : k\geq 0\}$. Solved by Chen and Ding [ChDi22]. See also [236]. OPEN Let$c_1,c_2>0$. Is it true that, for any sufficiently large$x$, there exist more than$c_1\log x$many consecutive primes$\leq x$such that the difference between any two is$>c_2$? This is well-known if$c_1\$ is sufficiently small.