MathDB

bright number is a sum of a perfect square and a perfect cube

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p4

January 27, 2020

number theoryPerfect Squareperfect cube

Problem Statement

A natural number is called bright if it is the sum of a perfect square and a perfect cube. Prove that if

r

and

s

are any two positive integers, then (a) there exist infinitely many positive integers

n

such that both

r+n

and

s+n

are bright, (b) there exist infinitely many positive integers

m

such that both rm and sm are bright.