The board / number theory / #455
Erdős Problem #455
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 `q : ℕ → ℕ` be a strictly increasing sequence of primes such that `q (n + 2) - q (n + 1) ≥ q (n + 1) - q n`. Must `lim q n / (n ^ 2) = ∞`?
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/455.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/455
Attempts
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\(\liminf q_n/n^2\ge1/(S-1)>0.543448148339064\), and exactly \(M(100000)=250\); the required limit remains open and reduces to linked-block sparsity (7).
PROVED