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Statement

Setting. $\mathcal U$ is the set of positive odd integers not of the form $p+2^k$ with $p$ prime and $k$ a positive integer (pp. 1--2); $\omega(m)$ is the number of distinct prime divisors of $m$.

Theorem 1.3 (p. 2). $\min m=11184810$ and $\min\omega(m)=7$, and $\omega(m)=7$ if and only if $m=11184810$, where both minima are taken over all infinite arithmetic progressions $\{mh+a:h=0,1,\ldots\}$ for which $\{mh+a:h=0,1,\ldots\}\setminus\mathcal U$ has asymptotic density zero.

Corollary 1.4 (p. 2). The same three conclusions hold with the minima taken over all infinite arithmetic progressions $\{mh+a:h=0,1,\ldots\}\subseteq\mathcal U$.

Here $11184810=2\cdot3\cdot5\cdot7\cdot13\cdot17\cdot241$. The paper notes (p. 3) that the value $\min m=11184810$ in Corollary 1.4 was obtained earlier by Chen, Dai and Li (arXiv:2402.06644) through a heavy calculation.

Source. Yong-Gao Chen, A conjecture of Erdős on $p+2^k$, arXiv:2312.04120v3 (2024). Labels and pages are those of arXiv v3: the statements on p. 2, the proofs in Section 4 (Lemmas 4.1--4.4 on pp. 17--23, the proofs of Theorem 1.3 and Corollary 1.4 on p. 23). The edition read is identified on the source card.

Read depth. Claims checked: the statements were read clause by clause on the printed pages, and the factorization of 11184810 was checked. The proof was read but not checked step by step; the paper omits the proof of Lemma 4.2, saying it can be verified directly. Nothing here is independently reviewed.

Proof pointer

Section 4. If $\{mh+a\}\setminus\mathcal U$ has density zero then $m$ is even and $a$ odd, and by a positive-proportion result of Sun (Lemma 4.4, p. 23) $(a-2^\ell,m)>1$ for every $\ell\ge1$. The odd primes $p\mid m$ with $a\equiv2^\ell\pmod p$ for some $\ell$ then form a "well constructed prime set": the residues $\ell\equiv a_i\pmod{r_{p_i}}$, with $r_p$ the order of 2 modulo $p$, cover the integers. Lemma 4.3 (p. 18, proof to p. 22), a case analysis on minimal covering systems using the necessary condition $\sum1/m_i\ge1$, Lemma 4.1 and the table of small orders in Lemma 4.2, shows such a set has at least six primes with product at least $5592405$, and exactly six if and only if the product is $5592405$. Attainment comes from Lemma 3.3 (p. 14): $\{11184810s+992077:s\ge0\}\subseteq\mathcal U$.

Dependencies

Lemma 4.4, quoted from Sun; Lemmas 4.1--4.3 on covering systems and orders of 2; Lemma 3.3 for the extremal progression.

Bears on

• Problem 16: the paper derives from Theorems 1.3 and 1.5 a third proof (p. 25) that the answer to the problem is no; see Theorem 3.1.