For problem #647 (Erdős–Selfridge: is there n>24 with max_{m<n}(m+τ(m)) ≤ n+2?), a computational update:
No n > 24 with R(n) = max_{m<n}(m+τ(m)) ≤ n+2 exists for n ≤ 1.0011×10^12.
Method: segmented sieve of τ(m) (numpy, block 10^7) maintaining the running record R(n); the known case n=24 is reproduced by the same detector as a sanity anchor (R(24)=26). Same pass and shard/overlap setup as the companion result for #385 — see https://github.com/Sanexxxx777/erdos-computational-bounds/tree/main/sieve647 (~18 core-hours total on an Apple M3, shared between both problems).
No claim on the prize question — no example was found in this range, so this is simply an extended verified bound, not progress toward $44.
For problem #647 (Erdős–Selfridge: is there n>24 with max_{m<n}(m+τ(m)) ≤ n+2?), a computational update:
No n > 24 with R(n) = max_{m<n}(m+τ(m)) ≤ n+2 exists for n ≤ 1.0011×10^12.
Method: segmented sieve of τ(m) (numpy, block 10^7) maintaining the running record R(n); the known case n=24 is reproduced by the same detector as a sanity anchor (R(24)=26). Same pass and shard/overlap setup as the companion result for #385 — see https://github.com/Sanexxxx777/erdos-computational-bounds/tree/main/sieve647 (~18 core-hours total on an Apple M3, shared between both problems).
No claim on the prize question — no example was found in this range, so this is simply an extended verified bound, not progress toward $44.