2-accessibility of primes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathcal P=\{2,3,5,\ldots\}$ be the positive primes. For $D\subset\mathbb N$, a $D$-diffsequence of length $k$ is a sequence of positive integers $x_1<\cdots<x_k$ with $x_{i+1}-x_i\in D$ for each $i$. The set $D$ is two-accessible if every coloring $c:\mathbb N\to\{0,1\}$ has monochromatic $D$-diffsequences of every finite length. The successive gaps need not be equal.

Statement recorded in the supplied source.

Is $\mathcal P$ two-accessible? Equivalently, for every $c:\mathbb N\to\{0,1\}$ and every integer $k\geq1$, must there exist $x_1<\cdots<x_k$ with all $c(x_i)$ equal and every difference $x_{i+1}-x_i$ prime? [S2]

Short English statement

Does every two-coloring of the positive integers have arbitrarily long monochromatic increasing sequences whose successive gaps are prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3042 (OPG-1797). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Oscar Quester, The Primes are 2-Accessible, arXiv:2606.00410v1 (2026), Definitions 1--2 and Theorem 2. The theorem gives an affirmative answer and proves that the accessibility degree of the primes is exactly two. https://arxiv.org/abs/2606.00410.