The board / number theory / #978
Erdős Problem #978
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).
Let `f ∈ ℤ[X]` be an irreducible polynomial with positive leading coefficient. Suppose that the degree `k` of `f` is larger than `2`, is not equal to a power of `2`, and `f n` has no fixed `(k - 1)`-th power divisors other than `1`. Then the set of `n` such that `f n` is `(k - 1)`-th power free has positive density, and this is proved in [Ho67].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/978.Newer public claim (6 Oct 2026): OpenAI's internal-model release claims, for this problem, n⁴+2 is squarefree with positive density; (d−2)-power-free values for d ≥ 4 — family 20 at github.com/openai/math, listed in its Lean catalogue. Bloom calls these claims, not endorsements; anything on this page that says the question is open was written before that release and has not re-checked it.
Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/978
Attempts
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Exactly 75,670 of the first 100,000 values n^4+2 are certified squarefree, the local density is rigorously enclosed in [0.756683702608, 0.756684005282], and the open problem is reduced to the unproved
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