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Claim. Lemma 2 of P. Erdős and A. Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, in Number Theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai 51 (1990), 45–91, digested on the card [[../library/arithmetic_functions/erdos_1990_distribution_values_certain_class_arithmetic_functions/_index|Erdős and Ivić 1990]]: the number of distinct prime factors of $p(1)p(2)\cdots p(n)$ tends to infinity with $n$. The lemma is attributed there to A. Schinzel, stated on p. 69 and proved on pp. 69–70 (result page [[../library/arithmetic_functions/erdos_1990_distribution_values_certain_class_arithmetic_functions/lemma_2|Lemma 2]]). The proof supposes that only the primes $q_1,\dots,q_r$ divide every $p(n)$ and reaches a contradiction from Tijdeman's theorem on the gaps between integers composed of a fixed finite set of primes ([[../library/arithmetic_functions/tijdeman_1973_integers_many_small_prime_factors/_index|Tijdeman 1973]]) together with the Hardy–Ramanujan asymptotic formula for $p(n)$. The site's commentary records that Schinzel first noted this argument in the Oberwolfach problem book, where Erdős asked the problem in 1986.

Covers. The first question of Problem 1106: $F(n)\to\infty$. The second question, whether $F(n)>n$ for all large $n$, is not addressed.

Depends on. No page of this wiki: the proof is in the cited paper.

Standing. Claimed. The paper appeared in the proceedings of the 1987 Budapest number theory conference, and no evidence that the volume was refereed is recorded, so refereed is not listed; the site labels the problem OPEN, so its commentary crediting the argument is not acceptance. The first question is settled by the refereed results on [[problems/arithmetic_functions/E1106/claims/1987_12_01_schinzel_wirsing|Schinzel and Wirsing's page]] and Ono's page. The page is named for Schinzel, whose argument the paper prints, and dated by the volume's year alone, since the record gives no finer date.