Perfect Cube
A perfect cube is an integer that is the product of some integer multiplied by itself three times.
Formula
n^3 = n \times n \times n
Definition
A perfect cube is a number you get when you multiply a whole number by itself and then by itself again, modeled as a cube-shaped arrangement. Formally, it is an integer of the form $n^3$ for some integer $n$; unlike perfect squares, perfect cubes can be negative, since $(-3)^3 = -27$, and the cube root of a perfect cube is always an integer. An integer $n$ is a perfect cube if and only if every exponent in its prime factorization is divisible by $3$; this connects to Waring's problem, which asks for the smallest $g(k)$ such that every positive integer is a sum of at most $g(k)$ perfect $k$-th powers, and for cubes $g(3) = 9$.
Example
$27$ is a perfect cube because $3 \times 3 \times 3 = 27$; others include $1, 8, 27, 64, 125, 216$. Is $-64$ a perfect cube? $(-4)^3 = -64$, yes; is $50$? Its cube root is about $3.68$, not an integer, so no. $23$ requires all $9$ cubes allowed by Waring's problem: $23 = 8+8+1+1+1+1+1+1+1$, one of the hardest cases to represent economically.
Key Insight
Picture a perfect cube as a 3D cube of small blocks: $27$ blocks can form a $3\times3\times3$ cube. Perfect cubes allow negative values because an odd number of negative factors gives a negative result, unlike perfect squares, which are always non-negative. Fermat's Last Theorem (Wiles, 1995) proves that $a^n + b^n = c^n$ has no positive integer solutions for $n \ge 3$; for $n=3$ this was first proved by Euler, and the broader problem of representing numbers as sums of cubes connects to deep results in analytic number theory.