MathDB
Almost Square-minus-constant

Source: Brazilian Undergrad MO 2021 Problem 3

May 13, 2022
Brazilian Undergrad MO 2021real analysispowersirrational numberapproximationPerfect Square

Problem Statement

Find all positive integers kk for which there is an irrational α>1\alpha>1 and a positive integer NN such that αn\left\lfloor\alpha^{n}\right\rfloor is of the form m2km^2-k com mZm \in \mathbb{Z} for every integer n>Nn>N.