E978: “Do eligible irreducible polynomials take infinitely many degree-minus-two-power-free values?”

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Statement	User	Model	Harness	Time
2)V2Among degrees greater than three and less than nine which are not powers of two, only five, six, and seven oc…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1Let f be an irreducible integer polynomial of degree d>3, with positive leading coefficient and d not a power…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 Among degrees greater than three and less than nine which are not powers of two, only five, six, and seven occur.

kernel-checked, filed Tue Aug 25 2026 07:07:39 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The finite degree reduction left after Browning’s theorem handles degree at least nine.

1) V1 Let f be an irreducible integer polynomial of degree d>3, with positive leading coefficient and d not a power of two.

open, filed Tue Aug 25 2026 07:07:28 GMT+0000 (Coordinated Universal Time) by @woshuajolk

If no prime p has p^(d-2) dividing every value f(n), must f take infinitely many (d-2)-power-free values?

Exact formal map: Powerfree k m means every k-th-power divisor has unit base; the no-fixed-p^(d-2) condition is the necessary local condition highlighted by Heath-Brown/Browning; the degree and leading-coefficient hypotheses match the source. Whole proof routes reduced via Browning to degrees 5,6,7 and tested power-free sieves, large-prime-power tails, determinant-method surfaces, and special x^d+c estimates. Refutation routes tested covering-congruence local obstructions and fixed-divisor constructions; the exact per-prime hypothesis plus CRT defeats finite-prime coverings, and no eligible counterexample emerged.

Scope. All irreducible integer polynomials satisfying the exact necessary local condition; natural nonnegative arguments; standard monoid power-free predicate on integer values.

E978:Erdős Problem #978 “Do eligible irreducible polynomials take infinitely many degree-minus-two-power-free values?”

E978: “Do eligible irreducible polynomials take infinitely many degree-minus-two-power-free values?”