Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 7

Problem 368

claims/: The 4 claim pages of Problem 368, one per claimant's result; the problem's standing derives from them.

Statement. How large is the largest prime factor of $n(n+1)$ ?

Status. Open, in the site's label (OPEN). Writing $F(n)$ for the prime in question, the site's commentary credits four refereed bounds, which the corpus accepts on their publication as partial claims, none determining the order of $F(n)$ : Pólya [Po18] proved $F(n)\to\infty$ (claim page), Mahler [Ma35] proved $F(n)\gg\log\log n$ (claim page), Schinzel [Sc67b] observed that $F(n)\le n^{O(1/\log\log\log n)}$ for infinitely many $n$ (claim page), and Pasten [Pa24b] proved $F(n)\gg(\log\log n)^2/\log\log\log n$ (claim page). The problem stays open: the site expects $F(n)\gg(\log n)^2$ for all $n$ , and Erdős [Er76d] conjectured that for every $\varepsilon>0$ there are infinitely many $n$ with $F(n)<(\log n)^{2+\varepsilon}$ .

Source. erdosproblems.com/368, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #368, https://www.erdosproblems.com/368.

References.

[Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.
[Ma35] Mahler, Kurt, Über den grössten Primteiler spezieller Polynome zweiten Grades. Archiv für math. og naturvid (1935).
[Pa24b] Pasten, Hector, The largest prime factor of $n^2+1$ and improvements on subexponential $ABC$ . Invent. Math. (2024), 373-385.
[Po18] Pólya, Georg, Zur arithmetischen Untersuchung der Polynome. Math. Z. (1918), 143-148.
[Sc67b] Schinzel, A., On two theorems of Gelfond and some of their applications. Acta Arith. (1967/68), 177-236.

Formalization. No formal-conjectures statement file exists for this problem. A file in Boris Alexeev's lean-proofs repository formalizes Pólya's qualitative statement and is linked on Pólya's claim page; the corpus has not built it.

Current assessment

The question, as the site states it, asks for the order of $F(n)$ , the largest prime factor of $n(n+1)$ ; it is an estimate problem with no yes-or-no answer, and the site records no parts. Four refereed results bound $F(n)$ , each the subject of an accepted partial claim. Pólya [Po18] proved $F(n)\to\infty$ , his Satz I applied to $x(x+1)$ , by reducing to Thue's theorem on binary forms; the Pólya and Mahler cards record that Størmer's earlier theory of the Pell equation $x^2-Dy^2=1$ already gives the same conclusion. Mahler [Ma35] proved $F(n)>(\log\log n)/(1+\varepsilon)$ for all large $n$ , from his theorem on the largest prime factor of $D_1x_0^2-A_0$ at $x_0=2n+1$ . Pasten [Pa24b] proved $F(n)\gg(\log\log n)^2/\log\log\log n$ , his Corollary 1.5 on the largest prime factor of $xy(x+y)$ for coprime $x<y$ taken at $x=1$ , $y=n$ , the best lower bound known. In the other direction Schinzel [Sc67b] proved that $F(n)\le n^{O(1/\log\log\log n)}$ for infinitely many $n$ , his Theorem 14 with $A=E=1$ , $r=2$ , $s=1$ , since $(2x+1)^2-1=4x(x+1)$ . Between these the truth is unknown: the site's commentary expects $F(n)\gg(\log n)^2$ for all $n$ , and Erdős [Er76d] conjectured that for every $\varepsilon>0$ infinitely many $n$ have $F(n)<(\log n)^{2+\varepsilon}$ . No source proves either, so the problem is open; the lower bounds supersede one another, and each is recorded as a partial claim because the site credits it.

The problem's thread (four comments as of 2026-10-07, no proof claim) holds the question whether Pasten's paper, whose title names $n^2+1$ , bears on $n(n+1)$ , Alexeev's answer of 2026-01-10 through Corollary 1.5, a comment of 2026-01-09 on Størmer's 1897 theorem, and Alexeev's post of 2026-02-17 linking the Lean file on Pólya's page. The community database records no formalization of the problem's statement. None of the four proofs is compiled in this wiki; the library cards for Mahler, Pasten, Schinzel and Pólya cover the statements.

Search scope: the site's problem page as exported, its thread as of 2026-10-07, the community database entry, the formal-conjectures tree (no statement file), the lean-proofs file and the four library cards; no forum proof claim and no OpenAI release item names this problem. No wider literature search was made, none being needed for refereed bounds the site credits.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (18)

Pólya: the largest prime factor of n(n+1) tends to infinity Mahler: F(n) exceeds log log n over 1 + epsilon for large n Schinzel: F(n) is at most n^{O(1/log log log n)} infinitely often Pasten: F(n) is at least a constant times (log log n)^2 over log log log n problems/arithmetic_functions/E0368/claims Arithmetic Functions Problem 373 arithmetic_functions/mahler_1935_uber_den_grossten_primteiler_spezieller Satz 1 (p. 15): the solutions of X^2 - D Y^2 = A whose Y has only prime factors dividing D Satz 2 (p. 20): each squarefree A has only finitely many singular D Satz 3 (p. 26) and the theorem of pp. 3–4: D_1 x_0^2 - A_0 has a prime factor above (log log x_0)/(1 + epsilon)arithmetic_functions/pasten_2024_largest_prime_factor_improvements_subexponential Corollary 1.5 (p. 3): the largest prime factor of xy(x+y) is at least kappa (log_2 y)^2/log_3 y Theorem 1.1 (p. 1): the largest prime factor of n^2+1 is at least a constant times (log_2 n)^2/log_3 n Theorem 1.4 (p. 2): subexponential ABC bounds of size exp(kappa sqrt((log R) log_2 R))arithmetic_functions/polya_1918_zur_arithmetischen_untersuchung_der_polynome Satz I (p. 144): the largest prime factor of a product of two essentially different linear factors tends to infinity arithmetic_functions/schinzel_nd_two_theorems_gelfond_applications

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