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Source. T. Tao, discussion comment on the Problem 251 page, 17:17 on 07 Oct 2025 (site clock): "By summation by parts, this is equivalent to the irrationality of $\sum_n\frac{p_{n+1}-p_n}{2^n}$." The comment states the equivalence in one line; the proof below is written out here and checked; it is elementary and complete.

Statement

Let $2=p_1<p_2<\cdots$ be the primes and $g_m=p_{m+1}-p_m$ the prime gaps. Both series below converge and

$$S:=\sum_{n\ge1}\frac{p_n}{2^n}=2+\sum_{m\ge1}\frac{g_m}{2^m},$$

so $S$ is irrational if and only if $\sum_{m\ge1}g_m2^{-m}$ is irrational.

Proof

Convergence: by Chebyshev's elementary bound $p_n\ll n\log n$, the terms $p_n2^{-n}$ are $O(n\log n\,2^{-n})$, so $\sum_np_n2^{-n}$ converges; the identity below then gives the convergence of the gap series, whose terms are positive.

For every $n\ge1$,

$$p_n=2+\sum_{m=1}^{n-1}g_m,$$

the sum being empty for $n=1$. Multiply by $2^{-n}$ and sum over $n\ge1$. All terms are nonnegative, so the order of summation may be exchanged:

$$\sum_{n\ge1}\frac{p_n}{2^n} =2\sum_{n\ge1}\frac1{2^n}+\sum_{m\ge1}g_m\sum_{n\ge m+1}\frac1{2^n} =2+\sum_{m\ge1}\frac{g_m}{2^m},$$

since $\sum_{n\ge1}2^{-n}=1$ and $\sum_{n\ge m+1}2^{-n}=2^{-m}$. Hence $\sum_{m\ge1}g_m2^{-m}=S-2$, and $S$ is rational exactly when the gap series is rational. $\square$

Numerically $S=3.67464396601\ldots$ (OEIS A098990), so the gap series equals $1.67464396601\ldots$.

Variants used by the 2026 manuscripts

For an integer base $B\ge2$ the same computation gives $\sum_{n\ge1}g_nB^{-n}=(B-1)\sum_{n\ge1}p_nB^{-n}-p_1$ (Ringer's equation (1)); with zero-based indexing $p_0=2$, $g_n=p_{n+1}-p_n$, it reads $\sum_{n\ge0}p_n2^{-n-1}=2+\sum_{n\ge0}g_n2^{-n-1}$ (Cook's form). Land works with the weighted tails $G_n=\sum_{j\ge0}g_{n+j}2^{-j-1}$, whose first value is $G_1=S-2$.

Relation to Problem 251

The identity changes nothing about the problem's status; it shows that the question is about the binary expansion of the dyadic series of prime gaps, where each gap $g_m\asymp\log m$ on average occupies several binary digits and the contributions overlap, so carries couple distant terms. Every 2026 manuscript on the problem starts from this form.

Bears on. #251 (an exact reformulation, not progress).