Erdos 1997 factor difference set integers

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Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. 79 (1997), no. 4, 353--359.

For an integer n the paper defines the factor-difference set D(n) = {|a-b| : n = ab} = {d_0 < d_1 < ... < d_k} and studies intersections of such sets and the gap structure of the sequence d_i. Proposition 3.1 shows that for any two distinct integers a, b only finitely many M have {a,b} contained in D(M), by turning {a,b} in D(M) into a factorization identity (x-y)(x+y) = (alpha-beta)(alpha+beta) and counting factorizations; Proposition 3.2 then constructs, for every k, integers N_1 < ... < N_k whose factor-difference sets share at least two common differences, using products of distinct odd primes. Conjecture 1 asks for k integers sharing at least k differences, supported by two explicit triples found by Barry Guiduli that share four differences. Section 4 shows the second smallest difference is large, Proposition 4.1 giving d_1(n)

= 2 n^{1/4}, which is used to determine the smallest difference d_0 = 16a+56 of the product a(a+1)...(a+7) of eight consecutive integers for a >= 5 and to produce infinitely many n with four divisors within c n^{1/4} of sqrt(n); gaps g_i = d_i - d_{i-1} are also discussed. The motivation, recounted in the introduction, is Erdős' question on placing n points in the plane with n^2/3 odd integral distances (later answered affirmatively by Piepmeyer), where the authors' attempted construction (points (±(2k_i+1)/2, 0) on the x-axis and (0, p_k) on the y-axis) needs integers 4p_k^2 whose factor-difference sets all contain the n integers 4k_i+2. The paper is thus the source for the factor-difference-set problems 885, 886 and 887.

Source: https://doi.org/10.4064/aa-79-4-353-359. The file's text layer carries no copyright or license line, and the publisher's record (https://www.impan.pl/get/doi/10.4064/aa-79-4-353-359, read 2026-10-02) offers the PDF under the link "Pobierz zgodnie z CC-BY" ("Free download under CC-BY license" on the English site), a Creative Commons Attribution license whose version the record does not name.

Bears on. #885, #886, #887

Results to transcribe.

• Proposition 3.1: For distinct integers a, b only finitely many M satisfy {a,b} ⊆ D(M).
• Proposition 3.2: For every k there exist N_1 < ... < N_k with |∩ D(N_i)| >= 2, built from products of distinct odd primes.
• Conjecture 1: For every k there exist N_1 < ... < N_k whose factor-difference sets share at least k differences.
• Proposition 4.1: The second smallest factor difference satisfies d_1(n) >= 2 n^{1/4}.