Erdős [Er58b] proved that $\sum \frac{p_n^k}{n!}$ is irrational for every $k\geq 1$.

In [Er88c] he further conjectures that $\sum \frac{p_n^k}{2^n}$ is irrational for every $k$, and that if $g_n\geq 2$ and $g_n=o(p_n)$ then\[\sum_{n=1}^\infty \frac{p_n}{g_1\cdots g_n}\]is irrational. (The example $g_n=p_n+1$ shows that some condition on the growth of the $g_n$ is necessary here.)

The decimal expansion of this sum is A098990 on the OEIS.

View the LaTeX source

This page was last edited 28 September 2025.

1 comment on this problem

• By summation by parts, this is equivalent to the irrationality of $\sum_n \frac{p_{n+1}-p_n}{2^n}$. It is possible that a sufficiently quantitative and uniform version of the prime tuples conjecture can resolve this problem, if it gives sufficient statistical control on the binary expansion of about $\log \log n$ consecutive prime gaps $p_{n+1}-p_n$ (which is usually of size $\asymp \log n$) to show that the binary expansion of $\sum_n \frac{p_{n+1}-p_n}{2^n}$ cannot be periodic. The theory of Shannon entropy may be helpful in this regard.

  TerenceTao — 17:17 on 07 Oct 2025

  👍 0 📝 0 🤖 0