Erdős Problem #884 — Is it true that, for any $n$, if $d_1<\cdots <d_t$ are the divisors of $n$, then $\sum_{1\leq i<j\leq t}\frac{1}{d_j-d_i} \ll 1+\sum_{1\leq i<t}\frac{1}{d_{i+1}-d_i},$ where the implied constant is…

Erdős Problem #884, attributed to [Er98, p. 177], asks whether a global reciprocal-distance sum over divisors is controlled by the corresponding sum over consecutive divisors.

September 2025 conditional disproof; subsequent unconditional claim

Terence Tao gave a disproof assuming the prime-tuples conjecture, using products of suitably shifted primes. Daniel Larsen subsequently supplied an unconditional disproof, reportedly by combining the construction over many scales so that the left-hand side becomes arbitrarily large. A Google DeepMind formalization records Tao’s conditional result, but its Lean theorem still contains sorry.

Current status (as of May 2026): The conjecture is recorded as disproved unconditionally by Daniel Larsen, while Tao’s earlier conditional disproof is documented; no independently verifiable published proof was found.