Hilbert's tenth problem for systems of diagonal quadratic forms, and B\"{u}chi's problem

TL;DR

This paper completes B"uchi's proof that there is no algorithm to decide the solvability of systems of diagonal quadratic equations over integers, by proving a specific property of sequences of squares related to B"uchi's problem.

Contribution

It proves a key property of squares satisfying a second difference condition, completing B"uchi's argument regarding the undecidability of certain quadratic systems.

Findings

No decision algorithm exists for the solubility of arbitrary diagonal quadratic systems.

Sequences of five squares with second differences equal to 2 are necessarily consecutive.

The result confirms B"uchi's conjecture and advances understanding of Hilbert's tenth problem.

Abstract

In this paper we complete B\"{u}chi's proof that there is no decision algorithm for the solubility in integers of arbitrary systems of diagonal quadratic form equations, by proving the assertion that whenever $x_1^2, \cdots, x_5^2$ are five squares such that the second differences satisfy \[x_{k+2}^2 - 2 x_{k+1}^2 + x_k^2 = 2\] for $k = 1,2,3$, then they must be consecutive. This answers a question of J.~Richard~B\"{u}chi.