Erdos 1952 greatest prime factor

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conjecture_p380: Records Erdős's 1952 expectation that the greatest prime factor of the product of an irreducible polynomial's first x values exceeds a constant times x to the degree, the bound asked in the second part of Problem 976.

theorem: Gives an iterated-logarithmic improvement to Nagell's $x\log x$ bound for the greatest prime factor of a product of polynomial values.

unproved_display_3: Records the stronger exponential-logarithmic bound that Erdős states but does not prove in the 1952 paper.

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Paul 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; DOI 10.1112/jlms/s1-27.3.379.

The abbreviation Zentralblatt 46,41 matches zbMATH's Zbl 0046.04102; MR 13,914a was not independently verified.

The copy read for this card is a six-page journal scan whose physical pp. 1--6 are printed pp. 379--384; the time it was downloaded is unknown. The scan is the Rényi Institute's Erdős archive copy of the offprint, which prints no copyright notice; the publisher's article pages could not be read on 2026-10-02 (HTTP 403), and this article's own Crossref record (DOI 10.1112/jlms/s1-27.3.379, read 2026-10-07) lists only Wiley's terms and conditions (http://onlinelibrary.wiley.com/termsAndConditions#vor) and its text-and-data-mining license (http://doi.wiley.com/10.1002/tdm_license_1.1), and no Creative Commons license, every other right reserved.

On physical p. 1 / printed p. 379, the paper introduces $P_x$ as the greatest prime factor of $\prod_{k=1}^x f(k)$ under the broad condition that the integer polynomial $f$ is not a product of linear factors with integer coefficients. It does not print a nonzero-product hypothesis or a convention for the greatest prime factor of zero. Taken literally, that broad condition permits, for example, $f(X)=(X-1)(X^2+1)$, for which the product is zero.

On the same page, equation (1) attributes the lower bound $P_x>c_1x\log x$ to Nagell. Erdős explicitly introduces his theorem as an improvement on that result, so the relevant predecessor scale is $x\log x$.

Immediately after equation (2), on the same page, the paper says that one may assume without loss of generality that $f$ is irreducible over $\mathbb Q$ and has degree greater than one. The native result therefore records this explicitly labeled safe specialization: for such an irreducible $f$ and all sufficiently large positive integers $x$, write $P_x$ for the greatest prime factor of $\prod_{k=1}^x f(k)$. The single theorem gives a constant $c_2=c_2(f)>0$ such that

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

The irreducible specialization has no integral root, so its running product is nonzero; nonconstant polynomial growth also makes the product's absolute value greater than one for all sufficiently large $x$. This is an editorially explicit domain qualification of the paper's immediate reduction, not an attribution of an unprinted hypothesis to its broader opening formulation. The broad wording's zero-product defect does not refute this specialization.

The paper's proof counts roots of $f$ modulo primes and invokes the prime ideal theorem. Lemma 1 uses selected semiprimes $a_i$ as divisors of values $f(t)$. Lemma 2 counts a separate family of inputs $u_i$ whose values avoid primes in a specified interval; Lemma 4 combines the two families. The remaining estimates compare prime-power parts with primes at most $x$. The six numbered lemmas and final contradiction occupy printed pp. 380--384. Lemma 6 on printed p. 384 is explicitly imported from Nagell; Erdős cites Nagell's 1922 paper, pp. 180--182, especially equation (7) on p. 182, for its proof. The local proof pointer identifies that dependency and the final use of equation (18).

After the irreducible reduction and the remark that equation (2) is far from best possible, Erdős writes the stronger display on printed p. 379:

$$P_x>x\exp\{(\log x)^{c_3}\}.$$

Printed p. 380 explicitly says that [[arithmetic_functions/erdos_1952_greatest_prime_factor/unproved_display_3|display (3) will not be proved in the paper]]. It therefore receives no proved-result credit here.

The same paragraph, on printed p. 380, adds that $P_x>c_4x^l$, with $l$ the degree of $f$, seems likely but, if true, must be very deep; the p. 380 remark records it as an unproved expectation.

This is a historical improvement on Nagell's bound relevant to Problem 976. The safe irreducible specialization applies to that problem's degree-at-least-two domain. The theorem for which this paper gives a proof and the stronger unproved display do not establish either universal fixed-power target on that page.

Source: https://users.renyi.hu/~p_erdos/1952-07.pdf.

Bears on. #976: the Theorem gives, for irreducible $f$ of degree greater than one, a lower bound $x^{1+o(1)}$ for the running product's greatest prime factor, short of the power gain $n^{1+c}$ the problem asks for; display (3) is a stronger assertion whose proof the paper withholds and which, with its unspecified $c_3>0$, gives no fixed power gain; the p. 380 remark expects, without proof, exactly the problem's degree-scale bound $F_f(n)\gg n^d$, $d$ the degree of $f$.

Results to transcribe.

• Theorem: the safe irreducible specialization of the bound proved in the paper, $P_x>x(\log x)^{c_2\log\log\log x}$.
• Display (3): the explicitly unproved assertion $P_x>x\exp\{(\log x)^{c_3}\}$ in the same safe specialization.
• Remark, p. 380: the unproved expectation $P_x>c_4x^l$, with $l$ the degree of $f$.

Living verification. Needs review. Physical pp. 1--6 / printed pp. 379--384 of the selected scan were read visually in full for the source identity, page map, opening convention, Nagell attribution and baseline, irreducible reduction, theorem, display (3), its no-proof qualification, the p. 380 remark on $P_x>c_4x^l$, and the two integer families and their roles in the proof map. This checks correspondence with the printed source, not every mathematical deduction. The external proofs of the prime ideal theorem and Nagell's Lemma 6 input were not read or reconstructed. No complete proof is supplied, reconstructed, or independently certified here.

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