The board / number theory / #450
Erdős Problem #450
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).
`m` has a divisor strictly between `n` and `2n`.
The number of integers strictly between `x` and `x + y` with a divisor in `(n, 2n)`.
Every window `(x, x+y)` has at most `ε y` integers with a divisor in `(n, 2n)`.
How large must \(y=y(\epsilon,n)\) be such that the number of integers in \((x,x+y)\) with a divisor in \((n,2n)\) is at most \(\epsilon y\)? The bound is required for every \(x\) and every window length at least \(y\), and \(y(\epsilon,n)\) is the least such threshold (or \(\infty\) if there is none). A **linear** scale \(y \le C(\epsilon) n\) is known to suffice for fixed \(\epsilon\) and all large \(n\) (see `erdos_450.linear_scale_suffices`).
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/450.Prizenone
Topicsnumber theory, divisors
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/450
Attempts
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Live page status is OPEN, but it lists 1 claimed proof—a full proof claimed by Colin Snyder (submitted 2026-07-15 01:30:36)—so the mandatory collision rule forbids an attempt.
SKIPPED