Existence of solutions to Erdős Problem 647 beyond the verified frontier

Determine whether there exists an integer n>24 satisfying max_{m<n}(m+τ(m))≤n+2 beyond the currently excluded range n≤9.17×10^18.

Background

Erdős Problem 647 asks whether any integer n>24 satisfies max_{m<n}(m+τ(m))≤n+2, where τ denotes the divisor-count function. The paper kernel-checks nonexistence only through 109, while cited computational searches extend the exclusion to approximately 9.17×1018 using a different trust base.

The authors explicitly state that their work is silent beyond 109 and does not resolve the problem globally. Thus, whether any solution exists beyond the current computational frontier remains unresolved; the paper only reports that heuristic considerations make such a solution appear very unlikely.

References

On the question itself the present work is silent beyond 109, and honesty requires repeating what the heuristics say: any solution, if one exists at all, lies beyond the current 9.17 × 1018 frontier, in ranges where the prime-tuple constraints price its existence very low.

— A Kernel-Checked Exclusion Certificate for Erdős Problem 647  (2608.17880 - Mian et al., 18 Aug 2026) in Section 6, “Limits and Prospects”