Question

find two consecutive intergers who cubes differ by 127

Answer 1

Call them x, y, x>y
\[x^3 - y^3 = (x-y)(x^2 + xy + y^2) \]
However, x = y+1 , so we actually have:
\[((y+1)-y)((y+1)^2+y^2+y(y+1)) = 3y^2+3y+1 = 127 \]