A characterization of B\"uchi's integer sequences of length 3

TL;DR

This paper provides a new characterization of B"uchi's integer sequences of length 3, focusing on sequences with constant second difference in their squares, and proposes a strategy for B"uchi's n Squares Problem.

Contribution

It introduces a novel characterization of generalized B"uchi sequences of length 3 over integers and offers a new approach to tackling B"uchi's n Squares Problem.

Findings

Characterization over integers differs from rational case.

Divisibility criteria are crucial for integer sequences.

Proposes a new strategy for B"uchi's n Squares Problem.

Abstract

We give a new characterization of generalized B\"uchi sequences (sequences whose sequence of squares has constant second difference $(a)$, for some fixed integer $a$) of length 3 over the integers and a strategy for attacking B\"uchi's n Squares Problem. Known characterizations of integer B\"uchi sequences of length 3 are actually characterizations over the rationals, plus some divisibility criterions that keep integer sequences.