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Statement

Setting (pp. 1--4). $\mathcal U$ is the set of positive odd integers not of the form $p+2^k$ with $p$ prime and $k$ a positive integer. In Problem 1.8 (p. 4), $A_i$ ($i\in I$) is the collection of all infinite arithmetic progressions of positive odd integers none of which is a prime plus a power of two, that is, all infinite progressions contained in $\mathcal U$.

Theorem 1.9 (p. 4). Let $a\in\mathcal U$. Then $a\in\bigcup_{i\in I}A_i$ if and only if there is an integer $m>1$ with $(a-2^k,m)>1$ for every positive integer $k$.

Corollary 1.10 (p. 4). $a\in\mathcal U\setminus\bigcup_{i\in I}A_i$ if and only if $a\in\mathcal U$ and $p(a-2^k)$ is unbounded (in $k$), where $p(n)$ is the least prime divisor of $n$ and $p(\pm1)=+\infty$ by convention. The paper says this is easily seen to be equivalent to Theorem 1.9.

Before the theorem (p. 4) the paper shows that $1,3,127$, each of the form $2^k-1$, lie in $\mathcal U$ but in no $A_i$. After it (p. 5) it records that $509203\in\mathcal U$ and that $\{11184810h+509203:h\ge0\}\subseteq\mathcal U$, so that the least element $e_1$ of $\bigcup_iA_i$ satisfies $149\le e_1\le509203$; Problems 1.12 and 1.13 ask for the exact least values.

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. 4, the proof in Section 5 on p. 26. The edition read is identified on the source card.

Read depth. Claims checked: the statements were read clause by clause on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Page 26. If $a$ lies in a progression $\{mh+b\}\subseteq\mathcal U$, Sun's positive-proportion result (Lemma 4.4, p. 23) gives $(b-2^k,m)>1$, hence $(a-2^k,m)>1$, for all $k\ge1$. Conversely, given such $m$, choose $k_0$ with $2^{k_0-1}>\max\{a,m\}$; any $n=p+2^k$ in $\{2^{k_0}mh+a\}$ would force $p\mid m$, and comparing residues modulo $2^{k_0}$ then gives $a=p+2^k$ or $a=p$, both impossible, so the progression lies in $\mathcal U$.

Dependencies

Lemma 4.4 (X.-G. Sun's positive-proportion theorem, p. 23).

Bears on

• Problem 16: context only. The theorem describes which elements of the problem's set are covered by infinite progressions contained in it; the paper's answer to the problem does not use it.