Erdos 1948 new questions distribution prime numbers

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lemma_p372: Erdős and Turán's unnumbered Lemma: for every constant A > 0 there are infinitely many k with p_k - p_{k-1} < p_{k+1} - p_k and p_k - p_{k-1} < A p_k^{1/2}, and infinitely many k with p_{k+1} - p_k < p_k - p_{k-1} and p_{k+1} - p_k < A p_k^{1/2}.

question_1: Erdős and Turán's question whether, for every fixed k, there are infinitely many n with p_{n+1} - p_n < p_{n+2} - p_{n+1} < ... < p_{n+k} - p_{n+k-1}; the case k = 3 is Erdős Problem 6.

theorem_1: Erdős and Turán's theorem that for every t the power mean ((p_{n-1}^t + p_{n+1}^t)/2)^{1/t} is larger than p_n for infinitely many n and smaller than p_n for infinitely many n, so neither the primes nor log p_n is convex or concave from some point on.

theorem_2: Erdős and Turán's theorem that for t < 1 an increasing integer sequence that is not an arithmetic progression from some point on, and satisfies a_k < k^2/4(1-t) - ck for every c once k is large, has ((a_{k-1}^t + a_{k+1}^t)/2)^{1/t} > a_k for infinitely many k; the growth condition is stated to be best possible, and only t = 0 is proved.

theorem_3: Erdős and Turán's companion to Theorem 2: for t > 1 an increasing integer sequence that is not convex from some point on, and satisfies the printed bound a_k < k^2/4(1-t) - ck for every c once k is large, has ((a_{k-1}^t + a_{k+1}^t)/2)^{1/t} < a_k for infinitely many k; the paper gives no proof.

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P. Erdős, P. Turán: On some new questions on the distribution of prime numbers, Bull. Amer. Math. Soc. 54 (1948), 371--378 (MR 9,498k; Zentralblatt 32,269). No notice is printed on the scanned pages (pp. 371--372 and 377--378 carry no copyright or license line); the journal's article page could not be read on 2026-10-02 (the Bulletin article address redirected to a page of the current volume), and the publisher's copyright policy page (https://www.ams.org/publications/authors/ctp, read 2026-10-02) states "AMS permits the noncommercial use of its copyrighted works for educational purposes only, such as to quote brief passages or to copy small portions of content for personal use in teaching or research" and names Creative Commons licenses only for five other AMS journals, not the Bulletin, and (as read on 2026-10-07) for authors' own postings of an accepted manuscript or draft, neither of which covers this publisher scan, every other right reserved.

Erdos and Turan ask whether log p_n is eventually convex and whether the primes themselves are eventually convex or concave, and answer both negatively. Theorem 1 (p. 372) proves that for every t both power-mean inequalities ((p_{n-1}^t + p_{n+1}^t)/2)^{1/t} > p_n and ((p_{m-1}^t + p_{m+1}^t)/2)^{1/t} < p_m have infinitely many solutions; the cases t = 0 and t = 1 give p_{n-1} p_{n+1} > p_n^2, p_{m-1} p_{m+1} < p_m^2, p_{n-1} + p_{n+1} > 2 p_n and p_{m-1} + p_{m+1} < 2 p_m. The proof is elementary and uses only pi(x) > c_1 x/log x, via a lemma producing infinitely many k with prescribed comparisons between consecutive gaps p_k - p_{k-1} and p_{k+1} - p_k. Theorems 2 and 3 (p. 374) give general statements for an increasing integer sequence a_k with a growth restriction a_k < k^2/(4(1-t)) - ck (for every c, once k is large; the print states this bound in Theorem 3 too, where t > 1 makes it negative): for t < 1 and a sequence that is not eventually an arithmetic progression, ((a_{k-1}^t + a_{k+1}^t)/2)^{1/t} > a_k infinitely often; for t > 1 and a sequence that is not eventually convex, the reverse inequality holds infinitely often. Only the case t = 0 of Theorem 2 is proved. Section 3 reproves the additive case t = 1 by a less elementary method (Page's prime number theorem for arithmetic progressions and Kuzmin's exponential sum bound) which the authors hope can attack the harder questions. Section 4 states without proof, by Brun's method, that for k <= n the power-mean difference ((p_{k-1}^t + p_{k+1}^t)/2)^{1/t} - p_k changes sign cn times, and that lim sup (p_{n+1} - p_n)/(p_n - p_{n-1}) > 1 and lim inf (p_{n+1} - p_n)/(p_n - p_{n-1}) < 1, and asks which linear forms take both signs infinitely often on consecutive primes. The paper ends (p. 378) with two questions: whether p_{n+1} - p_n < p_{n+2} - p_{n+1} < ... < p_{n+k} - p_{n+k-1} has infinitely many solutions for every fixed k (the case k = 3 is problem 6), and whether the number of k <= n with p_{k+1} - p_k > p_k - p_{k-1} is n/2 + o(n); there the authors say they can show this number lies between c_1 n and (1-c_1) n. In the print the proof of Theorem 2 at t = 0 (p. 375) says the inequality (13) "has finitely many solutions" [sic] where the argument that follows proves infinitely many.

Source: https://users.renyi.hu/~p_erdos/1948-05.pdf.

Bears on.

• #6: the case k = 3 of the paper's closing question (1) (p. 378) is the problem's statement; the paper poses it and does not answer it. Its Lemma (p. 372) and the case t = 1 of Theorem 1 give two consecutive increasing gaps infinitely often, not three.

Results.

• Theorem 1 (p. 372): For every t, ((p_{n-1}^t + p_{n+1}^t)/2)^{1/t} > p_n and ((p_{m-1}^t + p_{m+1}^t)/2)^{1/t} < p_m each have infinitely many solutions; in particular each of p_{n-1}p_{n+1} > p_n^2, p_{m-1}p_{m+1} < p_m^2, p_{n-1}+p_{n+1} > 2p_n and p_{m-1}+p_{m+1} < 2p_m has infinitely many solutions, so log p_n is neither eventually convex nor eventually concave.
• Lemma (p. 372): For any constant A > 0 there are infinitely many k with p_k - p_{k-1} < p_{k+1} - p_k and p_k - p_{k-1} < A p_k^{1/2}, and infinitely many k with p_{k+1} - p_k < p_k - p_{k-1} and p_{k+1} - p_k < A p_k^{1/2}; only pi(x) > c_1 x/log x is used.
• Theorem 2 (p. 374): For t < 1 and any increasing integer sequence that is not eventually an arithmetic progression and satisfies a_k < k^2/4(1-t) - ck for every c once k is large, ((a_{k-1}^t + a_{k+1}^t)/2)^{1/t} > a_k has infinitely many solutions; the paper proves only the case t = 0 and states that the growth condition is best possible.
• Theorem 3 (p. 374): For t > 1 and any increasing integer sequence that is not convex from some point on and satisfies the printed bound a_k < k^2/4(1-t) - ck for every c once k is large, ((a_{k-1}^t + a_{k+1}^t)/2)^{1/t} < a_k has infinitely many solutions; the paper gives no proof.
• Closing question (1) (p. 378): Asks whether, for every fixed k, there are infinitely many n with p_{n+1}-p_n < p_{n+2}-p_{n+1} < ... < p_{n+k}-p_{n+k-1}; the k = 3 case is Erdos problem 6. The page also records closing question (2) and the stated c_1 n to (1-c_1) n count.

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