Are there infinitely many balanced 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 $p_1=2<p_2=3<\cdots$ be all primes. A prime $p_j$ with $j\ge2$ is balanced (of order one) when $2p_j=p_{j-1}+p_{j+1}$, so the two adjacent prime gaps are equal.

Statement recorded in the supplied source.

Are there infinitely many indices $j\ge2$ with $2p_j=p_{j-1}+p_{j+1}$? [S2]

Short English statement

Do infinitely many primes lie exactly halfway between the immediately previous and next primes?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400144 (AMR-093-0144). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Are there infinitely many balanced primes. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A006562: Balanced primes (of order one): primes which are the average of the previous prime and the following prime., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A006/A006562.seq.