About

A cube number is the product of an integer multiplied by itself three times: n³ = n × n × n. Cube numbers grow rapidly. 10³ = 1000, but 100³ = 1,000,000. Misidentifying a perfect cube in engineering or volume calculations propagates cubic errors - literally. This tool generates verified cube number lists for any range from 1 to 10,000, formatted for reference or export.

The output includes the index n and its cube n³. This is useful for volume lookups (a cube with side length n cm has volume n³ cm³), combinatorics, number theory proofs, and quick mental math verification. Note: this tool computes integer cubes only. Fractional or negative cube roots require a different approach.

Formulas

The cube of an integer n is defined as:

n³ = n × n × n

Where n ∈ Z⁺ (positive integers). The inverse operation is the cube root:

n = √³x

A useful identity connects consecutive cubes to sums of odd numbers. The difference between consecutive cubes follows:

(n + 1)³ − n³ = 3n² + 3n + 1

Nicomachus' theorem states the sum of the first n cubes equals the square of the n-th triangular number:

n∑k=1 k³ = (n(n + 1)2)²

Where n is the upper bound of the range and k is the iteration index.

Reference Data

n	n³	Digit Count	Sum of Digits
1	1	1	1
2	8	1	8
3	27	2	9
4	64	2	10
5	125	3	8
6	216	3	9
7	343	3	10
8	512	3	8
9	729	3	18
10	1,000	4	1
11	1,331	4	8
12	1,728	4	18
13	2,197	4	19
14	2,744	4	17
15	3,375	4	18
16	4,096	4	19
17	4,913	4	17
18	5,832	4	18
19	6,859	4	28
20	8,000	4	8
25	15,625	5	19
30	27,000	5	9
40	64,000	5	10
50	125,000	6	8
75	421,875	6	27
100	1,000,000	7	1
150	3,375,000	7	18
200	8,000,000	7	8
500	125,000,000	9	8
1000	1,000,000,000	10	1

Frequently Asked Questions

Cube numbers grow at rate O(n³) versus O(n²) for squares. At n = 100, n² = 10,000 but n³ = 1,000,000 - a factor of 100× difference. By n = 1,000, the cube is 1,000,000,000 while the square is only 1,000,000. This cubic growth rate is why volume calculations are so sensitive to measurement error: a 1% error in side length produces approximately 3% error in volume.

A number that is both a perfect square and a perfect cube is a perfect sixth power: n⁶. The sequence is 1, 64, 729, 4096, 15625, 46656, 117649, ... corresponding to 1⁶, 2⁶, 3⁶, 4⁶, 5⁶, 6⁶, 7⁶. These are relatively rare. In the first 1000 cubes, only the first 10 (1⁶ through 10⁶) qualify.

Yes. Unlike squares (which can only end in 0, 1, 4, 5, 6, 9), cubes can end in any digit 0-9. The last digit of n³ equals the last digit of n itself. This property is unique to cubes and makes them identifiable: if a number ends in 7, its cube root (if integer) also ends in 7.

The tool supports n from 1 to 10,000. At n = 10,000, n³ = 1,000,000,000,000 (10¹²). JavaScript safely handles integers up to 2⁵³ − 1 (approximately 9 × 10¹⁵), so arithmetic remains exact within this range. Beyond n ≈ 2,097,152, floating-point precision loss begins affecting results.

Nicomachus' theorem (circa 100 CE) states that the sum of the first n cubes equals the square of the sum of the first n natural numbers: 1³ + 2³ + ... + n³ = (1 + 2 + ... + n)². For example, 1³ + 2³ + 3³ = 1 + 8 + 27 = 36 = 6² = (1+2+3)². This identity is used in series analysis and combinatorial proofs.

By Fermat's Last Theorem (proved by Andrew Wiles in 1995), no. There are no positive integer solutions to a³ + b³ = c³. However, some cubes can be expressed as sums of three or more cubes: 6³ = 3³ + 4³ + 5³ (216 = 27 + 64 + 125). The Hardy-Ramanujan number 1729 is the smallest number expressible as the sum of two cubes in two different ways: 1³ + 12³ = 9³ + 10³.