The board / number theory / #1061
Erdős Problem #1061
OPENdatabase updated 2025-09-28
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let `S x` count the number of **ordered** pairs of positive integers `(a, b)` with `a + b ≤ x` such that `σ(a) + σ(b) = σ(a + b)`, where `σ` is the sum of divisors function. In particular, `(a, b)` and `(b, a)` are counted separately; an unordered variant could be obtained by additionally requiring `a ≤ b`.
How many (ordered) solutions are there to `σ(a) + σ(b) = σ(a + b)` with `a + b ≤ x`? Is it true that this number is asymptotic to `c * x` for some constant `c > 0`?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/1061.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA110177
Original pageerdosproblems.com/1061
Attempts
submit one →Open — a recent Eric Li preprint claims more; we treat it as an external candidate, not ours.
NO PROGRESS