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Statement

Setting (pp. 1--2). $\mathcal P$ is the set of positive primes and $\mathbb N$ the set of positive integers. $\mathcal U$ is the set of positive odd integers that cannot be written as $p+2^k$ with $p\in\mathcal P$ and $k\in\mathbb N$; so $k\ge1$ throughout. The paper records $\mathcal U=\{1,3,127,149,251,331,\ldots\}$.

Theorem 1.1 (p. 2, quoted). "For any set $S$ of asymptotic density zero, $\mathcal U\cup S$ is not a union of finitely many infinite arithmetic progressions and a set of asymptotic density zero."

Corollary 1.2 (p. 2). Taking $S=\emptyset$: $\mathcal U$ itself is not a union of finitely many infinite arithmetic progressions and a set of asymptotic density zero.

Erdős's Conjecture A (p. 2), which the paper states as "The set $\mathcal U$ is the union of an infinite arithmetic progression of positive odd integers and a set of asymptotic density zero", is the case of one progression, so Corollary 1.2 refutes it.

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 statement on p. 2, the proof in Section 2, pp. 6--13. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement 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

Section 2, pp. 6--13, by contradiction. Suppose $\mathcal U\cup S=\bigcup_{i\le t}\{m_ih+a_i\}\cup W$ with $W$ of density zero; every $m_i$ is even, and $t\ge1$ by Erdős's progression in $\mathcal U$. The proof builds distinct primes $p_1,\ldots,p_s$ (prime factors of Fermat numbers $2^{2^{i-1}}+1$, of three numbers $(2^{3\cdot2^{2\ell-i}}+1)/(2^{2^{2\ell-i}}+1)$, and the remaining primes of $m_1\cdots m_t$) and integers $\alpha,a,c$ so that every $a-2^k$ has a prime factor among $p_1,\ldots,p_{\ell+3}$ while $p_{\ell+3}\nmid m_i$ and $p_i\nmid a-2^c$ for $i\ne\ell+3$. Then the progression $(p_1\cdots p_s)^\alpha h+a$ lies in $\mathcal U$ up to a density-zero set, so it meets some $\{m_1h+a_1\}$, and a Chinese-remainder shift at $p_{\ell+3}$ produces a positive proportion of $p+2^k$ in that progression, by Dirichlet's theorem, the second-moment bound $\sum_{n\le x}r(n)^2\ll x$ the paper cites from earlier work, and Cauchy--Schwarz (pp. 8--10). The construction of the primes is on pp. 10--13.

Dependencies

Dirichlet's theorem on primes in progressions; the bound $\sum_{n\le x}r(n)^2\ll x$ for the number $r(n)$ of representations $n=p+2^k$ (the paper's (2.11), cited from Chen and Sun and from Romanoff); Erdős's 1950 progression in $\mathcal U$.

Bears on

• Problem 16: the problem asks whether the odd integers not of the form $2^k+p$ are the union of an infinite arithmetic progression and a set of density $0$. Corollary 1.2, with $k\ge1$ as the paper fixes it, answers no, and Theorem 1.1 rules out finitely many progressions even after adding any density-zero set.