Erdos 1951 problems results elementary number theory

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equation_23: The sum of the squared gaps s_{i+1} - s_i between squarefree numbers with s_{i+1} at most x equals x times the sum of t^2 beta_t plus o(x), where beta_t is the density of the s_i with gap t; the paper sketches this case alpha = 2 of its moment asymptotic (23).

inequality_2: For the integers u_i of the form x^2 + y^2, infinitely many i have u_{i+1} - u_i greater than an absolute constant times log u_i / (log log u_i)^{1/2}, improving the log u_i / log log u_i bound Turán observed; Erdős deduces it from Theorem 1.

inequality_20: For the squarefree numbers s_i, infinitely many i have s_{i+1} - s_i greater than (1+o(1)) times pi^2/6 times log s_i / log log s_i; the paper adds that the matching upper bound (21) seems possible and records Roth's upper bound (22).

lemma_2: With g_t(x) the number of squarefree s_i < x followed by a gap of exactly t, the sum of g_l(x) over l > t is less than an absolute constant times x / (t^2 (log t)^2).

remark_p109: For the integers b_i divisible by no member of a sequence a_i, when the density of the b's exists the density of the b_i with gap t exists; when it is positive, gaps above a constant c_epsilon contribute less than epsilon x to the first moment; and an example with convergent sum of 1/a_i has an unbounded normalized (1+epsilon)-moment.

theorem_1: For primes p_i with divergent reciprocal sum and f(x) the sum of 1/p_i over p_i < x, the integers that are either prime to each p_i or divisible by its square have, infinitely often, a gap exceeding an absolute constant times e^{f(log v_i)} log v_i / log log v_i.

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P. Erdős: Some problems and results in elementary number theory, Publ. Math. Debrecen 2 (1951), 103--109 MR 13,627a; Zentralblatt 44,36.

Let p_1 < p_2 < ... be primes with sum 1/p_i divergent and let v_1 < v_2 < ... be the integers that for each i are either not divisible by p_i or are divisible by p_i^2; Theorem 1 (p. 103) shows that for infinitely many i one has v_{i+1} - v_i > c_4 e^{f(log v_i)} log v_i / log log v_i, where f(x) = sum_{p_i < x} 1/p_i. Taking the p_i to be the primes congruent to 3 mod 4 gives, for the integers u_i of the form x^2 + y^2, the gap bound u_{i+1} - u_i > c_2 log u_i / (log log u_i)^{1/2} for infinitely many i, improving Turan's observation of the weaker bound with log log u_i in the denominator; the paper also records the unproved conjecture that u_{i+1} - u_i = o(u_i^{1/4}), which would sharpen the bound u_{i+1} - u_i < c_1 u_i^{1/4} of Chowla and Bambah. The proof combines Brun's sieve bound for D_l(y), the number of integers below y divisible by none of p_1, ..., p_l, with a Chinese-remainder construction of a long run of non-v integers. For squarefree numbers s_i the same method yields s_{i+1} - s_i > (1+o(1))(pi^2/6) log s_i / log log s_i infinitely often (the print's (20) reads pi^3/6, a misprint for the pi^2/6 of the next sentence), Erdos suggests the matching upper bound with pi^2/6, notes Roth's bound s_{i+1} - s_i < s_i^{3/13} (log s_i)^{4/13+eps}, and sketches via Lemmas 1 and 2 the second-moment result sum_{s_{i+1} <= x} (s_{i+1}-s_i)^2 = x sum t^2 beta_t + o(x). This is the case alpha = 2 of the moment asymptotic (23), sum (s_{k+1}-s_k)^alpha = C_alpha x + o(x), which he hoped to prove for every alpha and states he can prove only for alpha below a constant between 2 and 3. The paper supplies the gap and moment results cited for Erdos problems 145, 208, 222 and 489.

Source: https://users.renyi.hu/~p_erdos/1951-13.pdf. No notice is printed in the file (pp. 103--104 and 108--109 carry no copyright or license line); the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, read 2026-10-02, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); the publisher's own file for DOI 10.5486/pmd.1951.2.2.04 (https://publi.math.unideb.hu/load_doi.php?pdoi=10_5486_PMD_1951_2_2_04, read 2026-10-02) is an image scan stating no term, and the Crossref record (read 2026-10-02) names the University of Debrecen as publisher and no license; the term is unstated.

Bears on. #145: the case alpha = 2 of (23) shows that the limit the problem asks about exists for alpha = 2; the paper states without proof that it can reach every alpha below a constant between 2 and 3. #208: the problem's second question is the upper bound (21), which the paper poses as seeming possible, and (20) shows its constant pi^2/6 could not be lowered; Roth's bound (22), recorded there, does not reach the first question. #222: (2) is a lower bound for infinitely many gaps between sums of two squares; the paper gives no upper bound of its own. #489: the case alpha = 2 of (23) is the problem's case A = prime squares, where the limit exists and is finite; the closing remarks on general sifted sequences settle no other case.

Results.

• Theorem 1 (p. 103, proof pp. 104--106): for primes p_i with divergent sum 1/p_i, the integers that are prime to each p_i or divisible by its square have, for infinitely many i, v_{i+1} - v_i > c_4 e^{f(log v_i)} log v_i / log log v_i (the print's (3) has v_{i-1} - v_i, a misprint).
• Inequality (2) (p. 103, deduced on p. 104): for u_i the integers of the form x^2 + y^2, u_{i+1} - u_i > c_2 log u_i / (log log u_i)^{1/2} for infinitely many i.
• Inequality (20) (p. 107): for s_i the squarefree numbers, s_{i+1} - s_i > (1+o(1))(pi^2/6) log s_i / log log s_i for infinitely many i (the print has pi^3/6, a misprint), with the possible upper bound (21) and Roth's bound (22).
• Lemma 2 (p. 107, proof pp. 107--108): the sum of g_l(x) over l > t is less than c_17 x / (t^2 (log t)^2), where g_l(x) counts s_i < x with s_{i+1} - s_i = l.
• (23) for alpha = 2 (stated p. 107, sketched pp. 108--109): sum_{s_{i+1} <= x} (s_{i+1} - s_i)^2 = x sum t^2 beta_t + o(x), with Lemma 1 (p. 107, cited from Mirsky) supplying the densities beta_t.
• Closing remarks (p. 109): gap densities for the integers divisible by no member of a sequence a_i, a first-moment tail bound, and an example with sum 1/a_i finite and unbounded (1+eps)-moment.

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