TITLE: $x^2+x+1$ is the cube of a prime.
QUESTION [14 upvotes]: Please help me find all natural numbers $x$ so that $x^2+x+1$ is the cube of a prime number.(Used in here)

REPLY [6 votes]: By Ljunggren (1942), the equation $x^2+x+1=y^3$ has only these integer solutions:
$(x,y)=(0,1)(-1,1)(18,7)(-19,7).$