Determine whether there exists an integer n>24 such that max_1≤ m<nbigl(m+tau(m)bigr)≤ n+2, where tau(m) denotes the number of positive divisors of m. Equivalently, the conjecture asserts that no such n exists.

Erdős problem 647

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Least-common-multiple formulation

Define $M(n,k)=\operatorname{lcm}(n+1,n+2,\ldots,n+k)$ . Determine whether there exist integers $m,n,k$ with $m\ge n+k$ such that $M(n,k)=M(m,k)$ .

source: Background supplied with the source

Erdős problem 647 asks whether the least common multiples of two separated blocks of consecutive integers can ever coincide: $M(n,k)=M(m,k)$ with $m\ge n+k$ . Erdős posed the problem in the context of his 1979 work; the main entry still lists it as open.

Known results

Thue–Siegel implies that, for fixed $k$ , only finitely many pairs $(m,n)$ satisfy the equality.
Erdős recorded the known examples $M(4,3)=M(13,2)$ and $M(3,4)=M(19,2)$ .
A related 2025 paper reports that the equivalent divisor-function formulation remains beyond its methods.

August 2026 kernel certificate

David Turean’s certificate checks a finite exclusion range end to end in Lean, strengthening confidence in that computational portion. It remains far below the prior non-kernel computational frontier and does not settle the conjecture. A January 2026 record also labels an AI-involving proof attempt incorrect.

Current status (as of August 2026): A finite exclusion range is kernel-verified, but the full conjecture and any possible exceptional cases remain open.

Erdős problem 647

Equivalent formulations 1

Least-common-multiple formulation

Primary source

Additional references

Progress summary

Solutions 0