Problem 240

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claims/: The 1 claim page of Problem 240, one per claimant's result; the problem's standing derives from them.

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Statement. Is there an infinite set of primes $P$ such that if $\{a_1<a_2<\cdots\}$ is the set of integers divisible only by primes in $P$ then $\lim a_{i+1}-a_i=\infty$?

Status. Proved, the site's label. The accepted claim is Tijdeman's Theorem 7 of 1973, refereed and credited by the site's curator.

Source. erdosproblems.com/240, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #240, https://www.erdosproblems.com/240.

References.

• [Po18] Pólya, Georg, Zur arithmetischen Untersuchung der Polynome. Math. Z. (1918), 143-148.
• [Ti73] Tijdeman, R., On integers with many small prime factors. Compositio Math. (1973), 319-330. Library home: tijdeman_1973_integers_many_small_prime_factors.

Formalization. None recorded on the site. Boris Alexeev's lean-proofs repository holds a Lean proof of the statement following Tijdeman's argument, recorded on the claim page; this corpus has not built it.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

• tijdeman_1973_integers_many_small_prime_factors
• tijdeman_1973_integers_many_small_prime_factors / theorem_3
• tijdeman_1973_integers_many_small_prime_factors / theorem_7