Is it possible for both a³ and (a+1)³ to exist within the interval between n² and (n+1)² ?

a & n are positive integers.

Let a^3=n^2+1

Then a=(n^2+1)^(1/3); and (a+1)^3=((n^2+1)^(1/3)+1)^3,

Which is n^2+(messy factors)+2. Ignore the mess.

But if n^2+2<2n+1, then, rearranging, (n-1)^2<0, which is not possible; so there are no solutions compliant with the requirements of the problem.

QED

Edited on March 11, 2015, 9:07 am

Posted by broll on 2015-03-11 09:02:39