Question 760026
{{{
n^3+(n+1)^3=(2n+1)(n^2-(n)(n+1)+(n+1)^2)
}}}

which gives us...
{{{
(2n+1)(n^2+n+1)
}}}

This number is to be 1 more than a perfect cube i.e.


{{{
m^3=2n^3+3n^2+3n=n(2n^3+3n+3)
}}}

We only have one equation so are algebra methods end here.



By trial and error n=1