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Source. Erdős (1952), unnumbered Theorem and equation (2), printed p. 379 (PDF, physical p. 1); the paper's proof ends on printed p. 384 (physical p. 6). Equation (1) on printed p. 379 credits Nagell with $P_x>c_1x\log x$, and Erdős explicitly introduces his theorem as an improvement on that bound.

Source convention and native domain. On physical p. 1 / printed p. 379, the paper first defines $P_x$ as the greatest prime factor of $\prod_{k=1}^x f(k)$ for an integer polynomial that is not a product of integer-linear factors. It prints no nonzero-product hypothesis and no greatest-prime-factor convention for zero. Thus the broad wording admits, for example, $f(X)=(X-1)(X^2+1)$, whose running product is zero.

Immediately after equation (2), the paper says that one may assume without loss of generality that $f$ is irreducible over $\mathbb Q$ and of degree greater than one. The following is the explicitly labeled safe specialization used natively; it does not silently add a hypothesis to the paper's broader opening wording. The broad wording's zero-product defect does not refute this specialization.

Safe irreducible specialization. Let $f\in\mathbb Z[X]$ be irreducible over $\mathbb Q$ with degree greater than one. For all sufficiently large positive integers $x$, let $P_x$ be the greatest prime factor of

$$\prod_{k=1}^x f(k).$$

Such an $f$ has no integral root, and nonconstant polynomial growth makes this product a nonzero integer of absolute value greater than one for all sufficiently large $x$. There is a constant $c_2=c_2(f)>0$ such that

$$P_x>x(\log x)^{c_2\log\log\log x}.$$

Proof pointer. Printed p. 380 defines the root-counting functions $\rho(k)$ and $\rho_x(k)$ and invokes the prime ideal theorem in equation (5). It then selects semiprimes $a_i\in(x/\log\log x,x)$ satisfying (7). Lemma 1 gives a lower bound for the number of integers $t\leq x$ for which some $a_i$ divides $f(t)$; its argument occupies printed pp. 380--382.

Printed p. 382 introduces a separate family of integers $u_i\in(x/\log x,x)$ for which $f(u_i)$ has no prime factor $p$ with $x\leq p\leq c_{13}x\log\log x$. It denotes their count by $U(x)$, and Lemma 2 gives a lower bound for $U(x)$.

On printed p. 383, under the assumed upper bound on $P_x$ that will lead to contradiction, the paper splits $f(k)=A_kB_k$, with $A_k$ formed from the full prime-power factors with primes at most $x$. Lemma 3 gives a lower bound for $A_{u_j}$. Lemma 4 combines Lemmas 1 and 2 to count the $u_j$ for which some $a_i$ divides $f(u_j)$, and Lemma 5 gives a stronger lower bound for $A_{u_j}$ on that subfamily. Lemma 6 on printed p. 384 states

$$\sum_{k=1}^x\log A_k<x\log x+c_{17}x.$$

Erdős credits this estimate to Nagell. The footnote cites the 1922 paper in Abhandlungen aus dem Mathematischen Seminar Hamburg, volume 1, pp. 179--194, locating the argument on pp. 180--182, especially equation (7) on p. 182; it says Nagell proves the estimate without stating it explicitly. This is an imported input, whose external proof is not included here.

The paper then combines Lemmas 2--5 in equation (18) on printed p. 384 and obtains its contradiction with Lemma 6 under the assumption that $P_x$ is smaller than the theorem's scale. This records the paper's proof map, not a complete reconstruction or an independent check of every deduction.

Relation to E976. The safe specialization applies to the irreducible degree-at-least-two case of Problem 976, but its extra factor is $x^{o(1)}$. It does not prove a fixed positive power gain.

Bears on. #976.

Living verification. Needs review. Physical pp. 1--6 / printed pp. 379--384 of the selected scan were read visually in full for the broad opening convention, missing zero convention, Nagell attribution and baseline, irreducible reduction, constant dependence, formula, standing large-$x$ convention, and the two integer families and their roles in the proof map above. This is a source-correspondence check; the external prime ideal theorem and Nagell proofs were not read or reconstructed. No complete proof is supplied, reconstructed, or independently certified here.