---

Statement

Let $f$ be a polynomial with integer coefficients, irreducible over $\mathbb Q$ and of degree $l>1$; this is the class to which printed p. 379 reduces the paper without loss of generality. Write $P_x$ for the greatest prime factor of $\prod_{k=1}^x f(k)$. The paper's convention on p. 379 makes $c_1,c_2,\ldots$ positive constants depending only on $f$ and takes $x$ sufficiently large.

At the top of printed p. 380, directly after withholding the proof of display (3), Erdős writes: "It seems likely that $P_x>c_4x^l$, but this if true must be very deep" (p. 380). In the corpus's words: he expects that for each such $f$ there is a positive constant $c_4=c_4(f)$ with

$$P_x>c_4x^l$$

for all sufficiently large $x$, and he gives no proof or argument for it.

This is a hedged expectation, not a stated theorem or a formally posed conjecture; the paper gives it no number or label, and this page names it by its page.

Source. P. Erdős, On the greatest prime factor of $\prod_{k=1}^x f(k)$, Journal of the London Mathematical Society 27 (1952), no. 3, 379--384; the sentence is on printed p. 380, with the reduction to irreducible $f$ of degree $l>1$ and the constant convention on printed p. 379. The edition read is identified on the source card.

Read depth. Claims checked: the sentence, the reduction and the constant convention were read clause by clause on the page images of printed pp. 379--380. A remark of this kind has no proof to check.

Dependencies

None. The remark sits beside the paper's Theorem, whose bound $x(\log x)^{c_2\log\log\log x}$ is $x^{1+o(1)}$ and so far below the expected scale $x^l$.

Bears on

• Problem 976: the remark is, for fixed irreducible $f$ of degree $l\geq2$, the bound asked for in the problem's second question, $F_f(n)\gg n^d$ with $d=l$. Since $l>1$, it would also answer the first question, $F_f(n)\gg n^{1+c}$, for that $f$. The paper offers it only as likely and gives no proof.