Büchi's problem

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.

A Büchi sequence of length $L\ge3$ is an integer tuple $(x_0,\ldots,x_{L-1})$ satisfying \[ x_{j+2}^2-2x_{j+1}^2+x_j^2=2\quad(0\le j\le L-3). \] It is trivial if there exists $a\in\mathbb Z$ with $x_j^2=(a+j)^2$ for every $0\le j<L$. Signs of the $x_j$ themselves are unrestricted; consecutive squares may decrease before increasing.

Statement recorded in the supplied source.

Does there exist an integer $M\ge3$ such that every Büchi sequence of every length $L\ge M$ is trivial? Equivalently, is there a uniform upper bound on the length of nontrivial integer Büchi sequences? [S2]

Short English statement

Must every sufficiently long integer-square sequence with second differences equal to two come from consecutive integers squared?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400001 (AMR-093-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, Büchi's problem, revision 1369143975 (2026) Lead and Statement of Büchi’s problem. https://en.wikipedia.org/w/index.php?title=B%C3%BCchi%27s+problem&oldid=1369143975.