Perfect Cube
ArithmeticA 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. It can be modeled as a cube-shaped arrangement.
Example
$27$ is a perfect cube because $3 \times 3 \times 3 = 27$. Perfect cubes: $1, 8, 27, 64, 125, 216$.
Key Insight
Picture a perfect cube as a 3D cube of small blocks. $27$ blocks can form a $3\times3\times3$ cube.
Definition
A perfect cube is an integer of the form $n^3$ for some integer $n$. Unlike perfect squares, perfect cubes can be negative: $(-3)^3 = -27$. The cube root of a perfect cube is always an integer.
Example
Is $-64$ a perfect cube? $(-4)^3 = -64$. Yes. Is $50$ a perfect cube? The cube root of $50 \approx 3.68$, not an integer. No.
Key Insight
Perfect cubes allow negative values because an odd number of negative factors gives a negative result. This contrasts with perfect squares, which are always non-negative.
Definition
An integer $n$ is a perfect cube iff every exponent in its prime factorization is divisible by $3$. Waring's problem asks: what is the smallest $g(k)$ such that every positive integer is a sum of at most $g(k)$ perfect $k$-th powers? For cubes, $g(3) = 9$ (every positive integer is a sum of at most $9$ cubes).
Example
$23$ requires $9$ cubes: $23 = 8+8+1+1+1+1+1+1+1 = 2^3+2^3+1^3 \times 7$. Most numbers need far fewer; $23$ is one of the hardest cases.
Key Insight
Fermat's Last Theorem (Wiles, 1995) proves that $a^n + b^n = c^n$ has no positive integer solutions for $n \ge 3$. For $n=3$ (cubes), this was first proved by Euler. The problem of representing numbers as sums of cubes connects to deep results in analytic number theory.