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Statement

Question (1) (p. 378, quoted). "Can the inequalities $p_{n+1}-p_n<p_{n+2}-p_{n+1}<\cdots<p_{n+k}-p_{n+k-1}$ have infinitely many solutions for every fixed $k$?"

The chain compares the $k$ consecutive gaps that start at $p_n$. The paper poses it as an open question. Its case $k=2$ follows from the first system (6) of the Lemma (p. 372) without its size condition.

The paper's second closing question (p. 378) asks whether the number of $k\le n$ with $p_{k+1}-p_k>p_k-p_{k-1}$ is $n/2+o(n)$; it says, without giving a proof, that it can show this number lies between $c_1n$ and $(1-c_1)n$.

Read depth. Claims checked: both closing questions and the stated count were read clause by clause on the page image of p. 378 of the print. Nothing here is independently reviewed.

Proof pointer

None: an open question in the paper.

Dependencies

None.

Source. P. Erdős and P. Turán, On some new questions on the distribution of prime numbers, Bull. Amer. Math. Soc. 54 (1948), 371--378; the edition read is named on the source card.

Bears on

• Problem 6: the case $k=3$ of the question, infinitely many $n$ with $d_n<d_{n+1}<d_{n+2}$ where $d_n=p_{n+1}-p_n$, is the problem's statement; the problem page lists the paper among its references.