Erdos 1955 remarks number theory hebrew

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conjecture_p48: The paper asks whether two sequences of integers up to n whose products a_i b_j are all distinct must satisfy xy < c_3 n^2/log n, and gives two constructions showing the bound would be best possible; the statement of Problem 490.

inequality_11: The count A(n) of integers up to n^2 that are a product of two integers not exceeding n is o(n^2), indeed o(n^2/(log n)^alpha) for some alpha > 0, proved from the Hardy–Ramanujan normal order of the number of prime factors.

main_theorem: Without the axiom of choice, Erdős builds a set of reals a_t of the power of the continuum such that any two of the integer sequences [a_t^n] are far apart in Hartman's sense, improving Hartman's aleph_1 such reals.

theorem_p47: For 2n distinct integers a_i in [1,4n] with complement b_j, some shift t gives at least n/2 solutions of a_i + t = b_j; the paper reports Scherk's (2 - sqrt 2)n and asks whether n can always be reached, the minimum overlap problem.

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P. Erdős: Some remarks on number theory (in Hebrew), Riveon Lematematika 9 (1955), 45--48 MR 17,460d. No notice is printed in the file (pp. 45 and 48, read as page images, carry only a received-date footnote and the author's affiliation); the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); the journal has no publisher page, so none was consulted, and no Crossref license is recorded; the term is unstated.

The paper is in Hebrew and carries an English summary on its last page (p. 48); its three numbered sections are unrelated. Section 1 (pp. 45--47) starts from Hartman, who called two increasing integer sequences far apart if |a_i - b_j| < A has only finitely many solutions for each A and had proved that there are aleph_1 reals a_alpha (1 <= alpha < Omega_1) with pairwise far-apart sequences [a_alpha^n]; Erdős builds, without using the axiom of choice, a set {a_t} of continuum many reals (a_t = e^{x_t} for 9/10 < t < 1, with x_t = sum 1/2^{[u_n t]} and u_n = 2^{3^n}, no ratio x_{t_1}/x_{t_2} being a Liouville number) whose sequences [a_t^n] are pairwise far apart, the key auxiliary lemma (p. 46) being a criterion (reduced approximations a_n/b_n with |alpha - a_n/b_n| < 1/(2 b_n^2) and b_{n+1} < b_n^{c_1}) guaranteeing that a number is not a Liouville number. He then asks (p. 47) for a field of real numbers of power c containing no irrational Liouville number, and says he could not even find such a ring; the English summary drops the word irrational. Section 2 (p. 47) shows that for any 2n distinct integers a_i in [1, 4n], with b_j the other 2n integers of (1,4n), some shift t gives at least n/2 solutions of a_i + t = b_j, reports Scherk's improvement to (2 - sqrt 2) n, and asks whether n is always reached, which a_i = n + i shows would be best possible. Section 3 (pp. 47--48) proves that the number A(n) of integers up to n^2 expressible as a product of two integers at most n is o(n^2), indeed o(n^2/(log n)^alpha) for some alpha > 0, using the Hardy–Ramanujan normal order log log n for the number of prime factors, and asks whether, when all products a_i b_j of a_1 < ... < a_x <= n and b_1 < ... < b_y <= n are distinct, xy < c_3 n^2 / log n, with two constructions showing this would be best possible.

Nothing in the paper touches the normalized prime gaps (p_{n+1}-p_n)/log n of Problem 5, whose site commentary cites this note under the key [Er55] for a theorem the note does not contain.

Source: https://users.renyi.hu/~p_erdos/1955-13.pdf.

Read status. Claims checked: the theorem of Section 1 with its lemma and question, the result and question of Section 2, inequality (11) and the question of Section 3 were read clause by clause on the page images of pp. 45--48, Hebrew text and English summary; the proofs were followed in outline.

Bears on. #36: with N = 2n, Section 2 proves the bound c >= 1/4 for partitions of {1, ..., 2N} with N even, reports Scherk's c >= 1 - 1/sqrt 2, and asks whether c = 1/2 is admissible (p. 47). #490: the question of p. 48 is the problem's statement, with the integers up to n/2 against the primes in (n/2, n) as the sharpness example; the paper states no bound toward it, though inequality (11) gives at once xy <= A(n) = o(n^2/(log n)^alpha), a consequence the paper does not draw. #896: the count F(A,B) is at most A(N), so inequality (11) gives max F(A,B) = o(N^2/(log N)^alpha) for some alpha > 0, weaker than the order the problem page records; the paper does not mention this quantity.

Results. The main theorem of Section 1 (p. 45, proof pp. 45--47, with the lemma of p. 46 and the question of p. 47); the shift theorem and question of Section 2 (p. 47); inequality (11) (p. 47, proof pp. 47--48); the distinct-products question (p. 48).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.