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Floor of powers of an irrational equals square minus constant

Source: Brazilian Mathematical Olympiad 2021, Level 3, Problem 3

February 7, 2022

number theoryfloor functionpowersirrational numberPerfect SquaresBrazilian Math OlympiadBrazil

Problem Statement

Find all positive integers

k

for which there is an irrational

\alpha>1

and a positive integer

N

such that

\left\lfloor\alpha^{n}\right\rfloor

is a perfect square minus

k

for every integer

n

with

n>N