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Statement

Theorem 2 (p. 374, quoted). "Let $a_1<a_2<\cdots$ be an infinite sequence of integers which do not form an arithmetic progression from a certain point on. Let $t<1$ and $a_k<k^2/4(1-t)-ck$, for every $c$ if $k$ is sufficiently large. Then

$$((a_{k-1}^t+a_{k+1}^t)/2)^{1/t}>a_k \qquad (11)$$

have infinitely many solutions."

The bound is printed as $k^2/4(1-t)$ and is read as $k^2/(4(1-t))$; at $t=0$ the paper writes it as $k^2/4$ (p. 375), and at $t=0$ the left side of (11) is the geometric mean $(a_{k-1}a_{k+1})^{1/2}$ (p. 375).

Sharpness (p. 374). The paper states that the inequalities in Theorems 2 and 3 are best possible in this sense: for every $c$ there is a sequence of integers $a_1<a_2<\cdots$ with $a_k<k^2/4(1-t)-ck$ for all $k$, not an arithmetic progression from any point on, for which (11) has only finitely many solutions, and the same holds for (12) of Theorem 3. For $t=0$ the paper's example (p. 375) is the sequence of all $n^2$ and $n(n+1)$ with an arbitrary finite set added.

Read depth. Claims checked: the statement, the sharpness claim and the proof of the case $t=0$ were read clause by clause on the page images of pp. 374--375 of the print. The general case is not proved in the paper. Nothing here is independently reviewed.

Proof pointer

Pp. 374--375, for $t=0$ only; the paper says the general case and Theorem 3 are similar but need slightly longer calculations, and gives neither. For $t=0$ the claim is that $a_{k-1}a_{k+1}>a_k^2$ (the paper's (13)) holds infinitely often under $a_k<k^2/4-ck$. The print says at this point that (13) "has finitely many solutions" [sic], but the argument that follows assumes the opposite of infinitely many solutions, which is what the theorem needs. If $a_{k-1}a_{k+1}\le a_k^2$ for all $k>k_0$, then with $x=a_{k+1}-a_k$ at an index where the gaps grow, $(a_k+x)^2\ge a_k(a_k+2x+1)$ forces $x^2\ge a_k$ (the paper's (14)); propagating this shows $(a_{k+1}-a_k)^2\ge a_k$ for all $k>k_0$, so each interval $[n^2,(n+1)^2)$ holds at most two terms for large $n$, which gives $a_k>k^2/4-ck$ for a large $c$ and contradicts the hypothesis.

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. The Remark on p. 374 says Theorem 1 follows from (5), the Lemma and Theorems 2 and 3.

Bears on

No Erdős problem in the corpus.