Is it possible for both
a^{3} and
(a+1)^{3} to exist within the interval between
n^{2} 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, (n1)^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 20150311 09:02:39 