Problem 370

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claims/: The 1 claim page of Problem 370, one per claimant's result; the problem's standing derives from them.

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Statement. Are there infinitely many $n$ such that the largest prime factor of $n$ is $<n^{1/2}$ and the largest prime factor of $n+1$ is $<(n+1)^{1/2}$?

Status. PROVED (LEAN), the site's label. The site records Steinerberger's trivial construction $n=m^2-1$ with $m$ and $m+1$ composite, and the formal-conjectures project links two Lean proofs of its statement; the site's curator suspects that the problem was misstated but offers no intended reading. The accepted claim is Steinerberger's construction.

Source. erdosproblems.com/370, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #370, https://www.erdosproblems.com/370.

Formalization. Statement in formal-conjectures (erdos_370, category research solved, as of its 2026-09-18 commit), which links two formal proofs: the lean-proofs file Erdos370.lean of 2025-11-24 and a proof of 2026-04-13 in a fork of formal-conjectures. This corpus has built and audited neither file.

Current assessment

Proved by a trivial construction; the intended problem is unknown. The site formulation above asks for infinitely many $n$ with $P(n)<n^{1/2}$ and $P(n+1)<(n+1)^{1/2}$, where $P$ is the largest prime factor; $n=m^2-1$ with $m$ and $m+1$ composite answers it, as the site records after Steinerberger, and two Lean proofs of the formal-conjectures statement are linked there. Erdős and Graham (p. 69) report Pomerance's observation that the system $P(n)>n^{e^{-1/2-\epsilon}}$, $P(n+1)>(n+1)^{e^{-1/2-\epsilon}}$ has solutions by density considerations, since integers with $P(n)>n^{e^{-1/2-\epsilon}}$ have density $1-\rho(e^{1/2+\epsilon})>1/2$, where $\rho$ is the Dickman function. The site's paraphrase, which puts the exponent $1/\sqrt e-\epsilon$ into the problem's own $<$ form, is mistaken: $n^{c}$-smooth integers have density $\rho(1/c)$, which exceeds $1/2$ only for $c>1/\sqrt e$, so the density argument for the $<$ form needs exponents above $1/\sqrt e$. The site's curator's remark that the problem was probably misstated is on the site; the nontrivial neighbors are Problems 369 and 928. No refereed source exists, and this corpus has built and audited neither Lean file. Sources checked: the site's page, its revision history (one revision, of 2025-10-20, with the remark present), the discussion thread (three comments, 2025-10-17 to 2025-12-21), the formal-conjectures statement file at its 2026-09-18 commit and both Lean files with their commit histories. The site lists no reference beyond Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathematique (1980), p. 69. The OpenAI release's manuscript on the joint Dickman law for consecutive integers does not name the problem; its library card OpenAI 2026 records, as a deduction made on the card and not in the manuscript, that its Theorem 1.1 at both exponents $1/2$ would give the problem's set positive lower density. That is a stronger form of a question the construction above already settles, so the release gets no claim page for this problem.

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.

• balog_1998_strings_consecutive_integers_no_large_prime_factors
• balog_1998_strings_consecutive_integers_no_large_prime_factors / theorem_1
• openai_2026_joint_dickman_law_consecutive_integers
• openai_2026_joint_dickman_law_consecutive_integers / theorem_1_1