Problem 1106

../

claims/: The 3 claim pages of Problem 1106, one per claimant's result; the problem's standing derives from them.

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Statement. Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of

$$\prod_{1\leq k\leq n}p(k).$$

Does $F(n)\to \infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$?

Status. Open, the site's label (OPEN, page last edited 16 November 2025). The first question is answered yes (Schinzel, with the proof in [ErIv90]; [ScWi87]; [On00]); the second, whether $F(n)>n$ for all large $n$, is open. The page lists the two parts as tends_to_infinity and exceeds_n.

Source. erdosproblems.com/1106, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1106, https://www.erdosproblems.com/1106.

References.

• [ErIv90] Erdős, Paul and Ivić, Aleksandar, [[../library/arithmetic_functions/erdos_1990_distribution_values_certain_class_arithmetic_functions/_index|The distribution of values of a certain class of arithmetic functions at consecutive integers]]. Number Theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai 51 (1990), 45-91.
• [Ob1] P. Erdős, Oberwolfach Mathematical Problems, Volume 1, Mathematisches Forschungsinstitut Oberwolfach; the site's source for the problem.
• [On00] [[../library/arithmetic_functions/ono_2000_distribution_partition_function_modulo_m/_index|Ono, Ken, Distribution of the partition function modulo $m$]]. Ann. of Math. (2) (2000), 293-307.
• [ScWi87] Schinzel, A. and Wirsing, E., Multiplicative properties of the partition function. Proc. Indian Acad. Sci. Math. Sci. 97 (1987), nos. 1-3, 297-303; doi:10.1007/BF02837831.
• [Ti73] Tijdeman, R., [[../library/arithmetic_functions/tijdeman_1973_integers_many_small_prime_factors/_index|On integers with many small prime factors]]. Compositio Math. (1973), 319-330.

Formalization. Statement in formal-conjectures.

Current assessment

First question answered yes; second question open. The problem asks whether the number $F(n)$ of distinct prime factors of $p(1)p(2)\cdots p(n)$ tends to infinity, and whether $F(n)>n$ for all large $n$; Erdős asked it at Oberwolfach in 1986 [Ob1]. Three results answer the first question. Schinzel's argument, from Tijdeman's gap theorem for integers composed of a fixed set of primes [Ti73] and the asymptotic formula for $p(n)$, is printed as Lemma 2 of [ErIv90] and recorded as a pending claim on its page, since the volume is a conference proceedings. Schinzel and Wirsing [ScWi87] give the rate $F(n)\gg\log n$, and Ono's theorem that every prime divides some $p(n)$ [On00] gives $F(n)\to\infty$ without a rate; both are refereed and are accepted partial claims on [[problems/arithmetic_functions/E1106/claims/1987_12_01_schinzel_wirsing|Schinzel and Wirsing's page]] and Ono's page. No result addresses the second question, so the problem's standing stays open. A comment on the discussion thread of 5 September 2026 reports, from tabulated values of $p(n)$, that $F(n)>n$ for $116\le n\le10000$; this is numerical data, not a claim. The formal-conjectures file states the first question as solved with answer(True), citing [ScWi87], and the second as open, both with sorry bodies.

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.

• erdos_1990_distribution_values_certain_class_arithmetic_functions
• erdos_1990_distribution_values_certain_class_arithmetic_functions / lemma_2
• ono_2000_distribution_partition_function_modulo_m
• ono_2000_distribution_partition_function_modulo_m / corollary_2
• ono_2000_distribution_partition_function_modulo_m / theorem_1
• tijdeman_1973_integers_many_small_prime_factors
• tijdeman_1973_integers_many_small_prime_factors / theorem_1
• tijdeman_1973_integers_many_small_prime_factors / theorem_2