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The factor-difference set of integers. (English) Zbl 0896.11008

The set \(D(n) = \{d:\;d=| a-b| ,\;n=ab\} = \{d_0 < d_1 < \cdots <d_k\}\) is studied in this paper. The motivating conjecture, still open, is that for all \(k>0\) there are \(k\) distinct integers \(N_1 < N_2 < \cdots <N_k\) with \( \left| \bigcap_{i=1}^k D(N_i)\right| \geq k\). This conjecture arose while trying to prove that one can place \(n\) points in the plane so that \(n^2/3\) of the distances determined by them are odd integers (this is now known to be true). The results proved are:
(i) For any two distinct positive integers \(a\) and \(b\) there are only finitely many \(n\) with \(\{a,b\} \subseteq D(n)\).
(ii) For each \(k\) there are \(N_1 < N_2 < \cdots <N_k\) with \(\left| \bigcap_{i=1}^k D(N_i)\right| \geq 2\).
(iii) \(d_1(n) \geq 2 n^{1/4}\) (the second smallest difference in \(D(n)\) is “large”).
(iv) Integers of the form \(N_a = a(a+1)\cdots(a+7)\), \(a\geq 5\), have \(d_i(N_a) \leq 16 N_a^{1/4}\) for \(i=0,1,2,3\).
Part (iv) shows that there exists an infinite set of integers with at least \(4\) “small” differences. The authors also show that the number \(4\) above cannot be increased using a similar method. Several open questions are pointed out.

MSC:

11B75 Other combinatorial number theory
11A51 Factorization; primality
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