The board / number theory / #955
Erdős Problem #955
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d\) be the sum of proper divisors function.
If \(A\subset \mathbb{N}\) has density \(0\) then \(s^{-1}(A)\) must also have density \(0\). A conjecture of Erdős, Granville, Pomerance, and Spiro [EGPS90].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 6 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/955.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/955
Attempts
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proved an elementary equivalence with uniform size-biased fibre tightness, isolating the exact missing lemma, and exactly certified the sharp finite concentration function through \(x=5{,}000{,}000\);
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