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Let denote the greatest prime factor of . Is it true that, for any two primes , there exists some integer such that and ?
Source: erdosproblems.com/649
An accepted solution exists. The statement is false.
Disproved; the site's label is DISPROVED (LEAN). The disproof is elementary and the site's remarks record two independent arguments, both accepted there: Tong's, for every prime infinitely many primes admit no such , and Sampaio's, the pair , . The site's Lean qualification refers to a Lean file posted in the site's thread. Its main theorem, the 2020 Romanian Master of Mathematics argument that infinitely many primes make a strange pair, is a third argument, accepted on its own claim page (Alexeev, 2026): this corpus built a later revision of the file and audited that theorem's statement. The file's formalizations of Tong's and Sampaio's results are links on their pages; their statements were not audited here, so they give no formalized evidence.