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Statement

Theorem 3 (p. 374, quoted). "Let $a_1<a_2<\cdots$ be an infinite sequence of integers which do not form a convex sequence from a certain point on (that is, $a_k-a_{k-1}>a_{k+1}-a_k$ has infinitely many solutions). Let $t>1$ and $a_k<k^2/4(1-t)-ck$ for every $c$ if $k$ is sufficiently large. Then

$$((a_{k-1}^t+a_{k+1}^t)/2)^{1/t}<a_k \qquad (12)$$

has infinitely many solutions."

As printed, for $t>1$ the bound $k^2/4(1-t)-ck$ is negative and tends to $-\infty$, while an increasing integer sequence satisfies $a_k\ge a_1+k-1$; so no sequence meets the hypothesis as printed. The paper gives no other form of the bound. Its sharpness statement (p. 374), recorded on the Theorem 2 page, says the same holds for (12).

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 374 of the print. The paper does not prove it. Nothing here is independently reviewed.

Proof pointer

None in the paper: it says (pp. 374--375) that the proof of Theorem 3 is similar to that of the case $t=0$ of Theorem 2 but needs slightly longer calculations.

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

No Erdős problem in the corpus.