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Source. Lemma 2, p. 69, proved on pp. 69--70, of Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Number Theory (Budapest, 1987), Colloq. Math. Soc. János Bolyai 51, North-Holland, Amsterdam (1990), 45--91, as identified on the source card. The paper attributes the lemma to A. Schinzel and thanks him for an unpublished result (p. 51).

Statement

Notation (pp. 45--46): $P(k)$ is the number of unrestricted partitions of $k$; $\omega(n)$ is the number of distinct prime factors of $n$.

Lemma 2 (A. Schinzel; p. 69, quoted). "If $\omega(n)$ is the number of distinct prime factors of $n$, then"

$$\lim_{n\to\infty}\omega\Bigl(\prod_{j=1}^{n}P(j)\Bigr)=\infty. \qquad(4.8)$$

Since $\omega\bigl(\prod_{j\le n}P(j)\bigr)$ is nondecreasing in $n$, the lemma says that infinitely many primes divide some value $P(j)$. It gives no rate of growth.

Proof pointer

Pp. 69--70. The proof uses an asymptotic formula for $P(n)$, displayed as (4.9) and cited to M. Knopp, Modular functions in analytic number theory (1970), p. 90, with main term $\frac{1}{4\sqrt3}\,e^{a\lambda_n}(n-1/24)^{-1}\bigl(1-\frac{1}{a\lambda_n}\bigr)$, where $\lambda_n=(n-1/24)^{1/2}$ and $a=\pi(2/3)^{1/2}$. Supposing that finitely many primes $q_1,\ldots,q_r$ account for the prime factors of every $P(n)$, $n\ge2$, it invokes R. Tijdeman's theorem (reference [30], On integers with many small prime factors, Compositio Math. 26 (1973), 319--330): there is a constant $C<0$ such that two integers $A$, $B$ composed only of $q_1,\ldots,q_r$ with $|A-B|\le A\log^CA$ are equal. It takes two sets of positive integers $a_1,\ldots,a_k$ and $b_1,\ldots,b_k$ whose $m$th power sums agree for $m=1,\ldots,[c/2]$ but not for $m=[c/2]+1$, and from (4.9) and Taylor expansion obtains that $A=\prod_jP(n+a_j)$ and $B=\prod_jP(n+b_j)$ have ratio $1+O(n^{-1/2-[c/2]})$ (4.10) and also $1+\Omega(n^{-1/2-[c/2]})$ (4.11). The first gives $A-B\ll A(\log A)^{-2[c/2]-1}$, hence $A=B$ by Tijdeman's theorem, against the second. The print names Tijdeman's constant $C$ and then works with $[c/2]$ without defining $c$ separately; the existence of the two sets of integers is asserted without proof or reference (both observations of this page).

Dependencies

The asymptotic formula (4.9) for $P(n)$ and Tijdeman's theorem, both cited. The lemma is used for Theorem 4. Read depth: claims checked; the statement was read clause by clause on p. 69 and the proof for its structure on pp. 69--70.

Bears on

• Problem 1106: with the problem's $p(n)$ written $P(n)$ here, the lemma states that $F(n)$, the number of distinct prime factors of $p(1)p(2)\cdots p(n)$, tends to infinity, which is the problem's first question. It gives no rate and does not address the second question, whether $F(n)>n$ for all large $n$.