For problem #385 (F(n) = max over composite m<n of m+p(m), p(m) the smallest prime factor of m — is F(n)>n for all sufficiently large n?), a computational update:
F(n) > n holds for all 10^4 < n ≤ 1.0011×10^12, except the two known values n = 267672 and n = 267680 (F(n) = n) — previously reported up to roughly 10^8 in the comments on Terry Tao's blog post on Siegel zeros (C.K.S., Aug 2024). This extends the verified range by a factor of about 10^4 and independently reproduces both known exceptional values (found by direct sieving, not by using the C.K.S. characterization of bad n).
Method: a segmented sieve computing τ(m) and the smallest prime factor spf(m) block-by-block (numpy, block 10^7), maintaining the running record F(n). The range was split into 8 shards with a 10^6 re-warm overlap (the running record is monotone, so each shard's local record catches up to the true one well within the overlap); shard boundaries were cross-checked against a single-process run bit-for-bit on a test range. ~18 core-hours on an Apple M3.
On the second question (does F(n)-n → ∞?): the minimum of F(n)-n grows monotonically across the swept range —
| range max |
min(F(n)-n) observed so far |
| 1.26e11 |
23,989 |
| 2.51e11 |
315,909 |
| 5.01e11 |
560,509 |
| 1.0011e12 |
868,399 |
consistent with F(n)-n → ∞.
Code and checkpoints: https://github.com/Sanexxxx777/erdos-computational-bounds/tree/main/sieve647
For problem #385 (F(n) = max over composite m<n of m+p(m), p(m) the smallest prime factor of m — is F(n)>n for all sufficiently large n?), a computational update:
F(n) > n holds for all 10^4 < n ≤ 1.0011×10^12, except the two known values n = 267672 and n = 267680 (F(n) = n) — previously reported up to roughly 10^8 in the comments on Terry Tao's blog post on Siegel zeros (C.K.S., Aug 2024). This extends the verified range by a factor of about 10^4 and independently reproduces both known exceptional values (found by direct sieving, not by using the C.K.S. characterization of bad n).
Method: a segmented sieve computing τ(m) and the smallest prime factor spf(m) block-by-block (numpy, block 10^7), maintaining the running record F(n). The range was split into 8 shards with a 10^6 re-warm overlap (the running record is monotone, so each shard's local record catches up to the true one well within the overlap); shard boundaries were cross-checked against a single-process run bit-for-bit on a test range. ~18 core-hours on an Apple M3.
On the second question (does F(n)-n → ∞?): the minimum of F(n)-n grows monotonically across the swept range —
consistent with F(n)-n → ∞.
Code and checkpoints: https://github.com/Sanexxxx777/erdos-computational-bounds/tree/main/sieve647