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Source. Printed p. 103, physical PDF p. 2, the paragraph beginning "I further proved", read on the page image; the bibliography on printed p. 109 (physical PDF p. 8) for the references.

Statement, verbatim

"I further proved that if $p_1<p_2<\ldots$ is the sequence of primes then $\sum_{n=1}^{\infty}p_n^k/n!$ is irrational for every $k$ [4]. I could not prove that $\sum p_n^k/2^n$ is irrational for every $k$. This is probably very difficult already for $k=1$. It seems reasonable to expect that if $g_n\ge2$, $g_n/p_n\to0$ then

$$\sum_{n=1}^{\infty}p_n/g_1\ldots g_n\qquad(2)$$

is irrational, but I can prove the irrationality of (2) only under much more restrictive conditions; $g_n=p_n+1$ shows that some growth condition is needed for the irrationality of (2)."

Reference [4] is "P. Erdős, Sur certaines series a valeur irrationnelle, Enseignement Math. 4 (1958), 93--100", filed as erdos_1958_sur_certaines_series_valeur_irrationnelle_french.

Notes

• The first sentence overstates the 1958 paper. That paper proves $\sum p_n/n!$ irrational and asserts the cases $k\ge2$ without proof (its main theorem). The first published proof for $k\ge2$ is Schlage-Puchta 2007, Theorem 3. Nothing here attributes the cases $k\ge2$ to Erdős.
• The second and third sentences are problem 251 for $k=1$; the site's page states the expectation for every $k$ as a remark citing this page.
• The expectation about (2) assumes only $g_n\ge2$ and $g_n=o(p_n)$, with no monotonicity. The paper does not say which "much more restrictive conditions" it means. Two theorems that prove (2) irrational under such conditions are in papers it cites: the 1958 section 3 theorem (its [4]: nondecreasing $q_n$ with the growth hypothesis (5), the sum rational only when $q_n=qp_n+1$ eventually) and Erdős–Straus 1974, Theorem 3.1 (its [3]: monotone $a_n$ with $p_n=o(a_n^2)$ and $\liminf a_n/p_n=0$). The 2026 Kovač note claims an explicit sequence $g_n\ge2$ with $g_n=o(p_n)$, not monotone, whose sum (2) is exactly $1$; it is a claimed counterexample to the expectation as stated here, non-refereed, with no independent review filed in this library. The monotone theorems (Erdős 1958 section 3; Hančl–Tijdeman 2004, Theorem 5.1 and Theorem 6.1) are not affected.
• The example $g_n=p_n+1$ gives sum $1$ by telescoping ($p_n/(g_1\cdots g_n)=1/(g_1\cdots g_{n-1})-1/(g_1\cdots g_n)$); it is the case $q=1$ of the rational family $q_n=qp_n+1$ in the 1958 theorem.

Bears on. #251.