Finding Squares Between Cubes in Elementary Number Theory

[tara123]

Im really good at number theory but how to show this statement has me stumped!

"Show that among the positive integers greater than or equal to 8, between any two cubes there are at least 2 squares"

 

[Billy Bob]

Can you find 2 squares between n^3 and (n+1)^3?

 

[tara123]

yah if u allow for the restriction of n>=8
if u have 8^3=512 and 9^3=729
then there's 23^2=529 and 24^2=576 both of which are between the cubes..

 

[Billy Bob]

You have found two squares between the two particular cubes 8^3 and 9^3, but what about between two generic cubes n^3 and (n+1)^3, where n is arbitrary (and >1).

You can do it by showing it is not possible to have two cubes between m^2 and (m+2)^2. That is, assume m^2 < n^3 and (n+1)^3 < (m+2)^2 and deduce a contradiction.