Are there infinitely many $n$ with $n^4+2$ squarefree? Power-free values of polynomials (Erdős #978)
Statement
Let $f\in\mathbb{Z}[x]$ be irreducible of degree $k>2$ with positive leading coefficient, and suppose $k\neq 2^l$ for any $l\ge 1$. (i) Does the set of integers $n\ge 1$ for which $f(n)$ is $(k-1)$-power-free have positive density? (ii) If $k>3$, and for every prime $p$ there is some $n$ with $p^{k-2}\nmid f(n)$, are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free? As a concrete instance of (ii): does $n^4+2$ represent infinitely many squarefree numbers? (An integer is $j$-power-free if no prime $p$ satisfies $p^{j}\mid$ it.)
Acceptance. FULLY RESOLVES: a proof (machine-checkable preferred, else a complete written proof) settling question (ii) in the open range — in particular an unconditional proof that $n^4+2$ takes infinitely many squarefree values (or, generally, that every eligible irreducible $f$ of degree $4\le k\le 8$ has infinitely many $(k-2)$-power-free values under the stated local hypothesis), or a proof that it fails for some eligible $f$. ADVANCES (proof required): (a) resolve a single new degree $k\in\{5,6,7,8\}$ of question (ii), lowering the threshold below the value $9$ of Browning stated in the background; (b) an unconditional infinitude (or density) result for squarefree values of a specific quartic such as $n^4+2$; or (c) a conditional resolution under a clearly-flagged hypothesis (e.g. abc) that is more explicit or quantitative than what is known. Deliver the proof or formalization. Empirical squarefree-density computations may support but do not by themselves close this proof-shaped problem.
Background
Posed by Erdős [Er65b, p.219] and [Er81h, p.178]; listed as open on erdosproblems.com/978 (fetched 2026-07-21, status 'open'). Progress: Erdős [Er53] proved there are infinitely many $n$ with $f(n)$ being $(k-1)$-power-free, except in the degenerate case $k=2^l$ with $2^{k-1}\mid f(n)$ for all $n$ (possible, e.g. $f(x)=k!(\binom{x}{k}+1)$). Hooley [Ho67] fully settled question (i), with a precise asymptotic for the count of such $n\le x$. For question (ii), Heath-Brown [He06] proved 'yes' for $k\ge 10$ and Browning [Br11] extended this to $k\ge 9$ (with an asymptotic formula), under the necessary local hypothesis that no prime $p$ has $p^{k-2}\mid f(n)$ for all $n$. The cases $4\le k\le 8$ of (ii) remain open; in particular whether $n^4+2$ (the $k=4$, squarefree case) takes infinitely many squarefree values is unresolved — infinitude of squarefree values of irreducible quartics is not known unconditionally, though it follows from the abc conjecture. Erdős [Er65b] also raised the 'intractable' questions of whether $2^n\pm 1$ or $n!\pm 1$ take infinitely many $k$th-power-free values (cf. erdosproblems.com/936). A formalisation exists in the DeepMind formal-conjectures library. Attacker's tool: sieve out square (and higher-power) divisors of $f(n)$ over long ranges to compute the empirical density of squarefree values of $n^4+2$ and calibrate the conjectural constant, alongside the square-sieve / abc-conditional analytic machinery of Heath-Brown and Browning.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #978 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #978 (formal-conjectures) | website |
Investigations · 0
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