The board / number theory / #893
Erdős Problem #893
OPENdatabase updated 2025-08-31
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).
Definition of function \(f(n) := \sum_{1\leq k\leq n}\tau(2^k-1)\). Here \(\tau\) is the divisor counting function, which is `σ 0` in mathlib.
Does the limit \(\lim_{n\to\infty} \frac{f(2n)}{f(n)}\) tend to infinity? (Other finite limits have been ruled out by [KoLu25], see below)
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/893.Prizenone
Topicsnumber theory, divisors
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA046801
Original pageerdosproblems.com/893
Attempts
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proved the uniform bound f(2n)/f(n)>2 and a Bang–Zsigmondy weighted reduction, with an independent exact certificate through n=138; full divergence remains open.
PARTIAL