The Largest Prime Factor of an Irreducible Cubic Polynomial

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main_theorem: Gives a positive-density pointwise large-prime-factor result and its running-product consequence for monic irreducible cubics.

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Ivan Ermoshin, The Largest Prime Factor of an Irreducible Cubic Polynomial, arXiv:2602.03642v3, updated 12 June 2026.

Copy read. The copy read for this card is the arXiv v3 PDF, which has 28 physical pages. The exact reviewer download time is unknown. That copy establishes a preprint version, not journal publication or independent acceptance. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2602.03642), every other right reserved.

For an integer polynomial $f$, the paper defines

$$P(x,f)=P^+\!\left(\prod_{n\leq x}f(n)\right).$$

The unnumbered main theorem on p. 3 (printed and physical page numbers agree) states that for every monic irreducible cubic $f\in\mathbb Z[X]$ there is a constant $c>0$ such that a positive proportion of the integers $n\in[x,2x]$ have a prime factor of $f(n)$ exceeding $x^{1+c}$; in particular $P(x,f)\gg x^{1+c}$. The abstract (p. 1) says the exponent depends on the polynomial, and no explicit value is given; the corpus writes $c_f$ and $\gg_f$ for that dependence.

This is unconditional positive-power progress for the full class of monic irreducible cubics. The paper recalls (p. 3) that Heath-Brown asked whether his result for $x^3+2$ generalizes to arbitrary cubic polynomials; the theorem does so for the monic irreducible ones. It covers only that class among the irreducible polynomials of degree at least two in Problem 976, and it does not reach the separate $x^3$ target there.

Sections 1.3--5 (pp. 3--27) outline and execute a generalization of Heath-Brown's cubic method. The argument expresses $f(n)$ as a norm in the cubic field generated by a root of $f$, constructs a positive-density set with an excess small-prime logarithmic contribution (Lemma 2, p. 6), separates a main term $S_0$ from an error $S_1$ (p. 7) and reduces the theorem to $S_0\gg1$ and $S_1=o(X)$ (Lemma 3, p. 8). Section 4 uses a $q$-van der Corput estimate to get $S_1=o(X)$ for $\delta\leq10^{-3}$ (p. 22), and Section 5 proves $S_0\gg1$ through ideal and fundamental-domain counting (p. 27). This is a proof map only.

Source: https://arxiv.org/abs/2602.03642.

Read status. Claims checked: the main theorem, the definition of $P(x,f)$ (p. 2) and Lemmas 2 and 3 were read on the page images of pp. 1--8, 15--17 and 21--28. The estimates of Sections 2--5 were not re-derived, and nothing here is independently reviewed.

Bears on. #976: for each monic irreducible cubic $f$ the main theorem gives $F_f(n)\gg_f n^{1+c_f}$ with some $c_f>0$, the problem's first question for that class only; it gives no exponent uniform in $f$, no result for other irreducible polynomials and no bound $F_f(n)\gg n^3$.

Result. Main theorem (unnumbered, p. 3).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.