Can a negative number be a perfect cube?

Yes. A negative integer is a perfect cube if its absolute value is a perfect cube. For example, -8 is a perfect cube because (-2) cubed equals -8, and -125 is a perfect cube because (-5) cubed equals -125. Odd roots of negative numbers are always real and negative.

How many perfect cubes are there between 1 and 1,000,000?

There are exactly 100 perfect cubes from 1 to 1,000,000 (from 1 cubed to 100 cubed, since 100 cubed equals 1,000,000). In general, the count of perfect cubes from 1 to N is the floor of the cube root of N. The cube root of 1,000,000 is exactly 100.

What is the formula to check if n is a perfect cube?

Let r be the nearest integer to the cube root of n. Compute r cubed. If r cubed equals n, then n is a perfect cube. In code: r = round(cbrt(n)), check r times r times r equals n. This avoids floating-point errors that can arise from directly comparing a computed cube root to its integer part.

Are there infinitely many perfect cubes?

Yes. For every positive integer k, k cubed is a perfect cube. Since the integers go on forever, so do the perfect cubes. However, perfect cubes become increasingly sparse as numbers grow larger. Up to N, there are only about the cube root of N perfect cubes, so they thin out rapidly compared to, say, even numbers.

Perfect Cube Calculator

Q: What is a perfect cube?

A perfect cube is an integer that equals some integer multiplied by itself three times. For example, 27 is a perfect cube because 3 times 3 times 3 equals 27. Equivalently, a perfect cube is any integer n for which the cube root of n is also an integer. The sequence of positive perfect cubes starts at 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Q: How do you check if a number is a perfect cube?

Compute the cube root of the number. If the result is a whole number (integer), then the original number is a perfect cube. For example, the cube root of 64 is 4 (an integer), so 64 is a perfect cube. The cube root of 70 is approximately 4.121, which is not an integer, so 70 is not a perfect cube.

Q: What are the perfect cubes from 1 to 1000?

The perfect cubes from 1 to 1000 are 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. There are exactly 10 of them. Their cube roots are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 respectively.

Q: What is the difference between a perfect square and a perfect cube?

A perfect square is an integer n where the square root of n is an integer: for example, 9 = 3 squared. A perfect cube is an integer n where the cube root of n is an integer: for example, 27 = 3 cubed. Some numbers are both: 64 = 8 squared and 4 cubed, 729 = 27 squared and 9 cubed. Numbers that are both are called perfect sixth powers.

Instantly check if a number is a perfect cube and find all perfect cubes up to any limit.

³ What is a Perfect Cube?

A perfect cube is an integer that can be written as the cube of another integer. More precisely, a positive integer n is a perfect cube if there exists a positive integer k such that k × k × k equals n. The first few perfect cubes are 1 (1³), 8 (2³), 27 (3³), 64 (4³), 125 (5³), 216 (6³), 343 (7³), 512 (8³), 729 (9³), and 1000 (10³). A number is a perfect cube if and only if its cube root is an exact integer with no fractional part.

Perfect cubes arise naturally in many areas. In geometry, the volume of a cube with integer side length is always a perfect cube: a cube with side 5 cm has volume 125 cm³. In number theory, perfect cubes play a role in Fermat's Last Theorem, which states that there are no positive integer solutions to a³ + b³ = c³ (proved by Andrew Wiles in 1995). In computer science, memory sizes and 3D grid dimensions often involve powers of 2 which are perfect cubes: 8, 512, and 134,217,728 are all perfect cubes of powers of 2. In algebra, the difference of cubes formula a³ − b³ = (a − b)(a² + ab + b²) and the sum of cubes formula a³ + b³ = (a + b)(a² − ab + b²) apply whenever a and b are integers.

An important distinction is between perfect cubes and perfect squares. A perfect square has a square root that is an integer (1, 4, 9, 16, 25, ...). A perfect cube has a cube root that is an integer (1, 8, 27, 64, 125, ...). Some numbers are both, called perfect sixth powers: 64 = 2&sup6;, 729 = 3&sup6;, 4096 = 4&sup6;. Additionally, negative integers can also be perfect cubes: −8 is a perfect cube because (−2)³ = −8, a property that does not hold for perfect squares (which are always non-negative).

This calculator provides two modes. In Check mode you enter any integer and instantly learn whether it is a perfect cube, what its exact or approximate cube root is, and which perfect cubes immediately precede and follow it. In List mode you set an upper limit and the calculator generates a complete table of all perfect cubes from 1 to that limit with their cube roots.

📐 Formula

n is a perfect cube ⇔ ³√n ∈ ℤ

n = the integer to test

³√n = cube root of n; must be an exact integer for n to be a perfect cube

Check method: compute k = round(³√n), then verify k³ = n

Negative cubes: n is a perfect cube if and only if −n is also a perfect cube; ³√(−n) = −³√n

Count up to N: number of perfect cubes from 1 to N = ⌊³√N⌋

Example: Is 343 a perfect cube? ³√343 = 7, and 7³ = 343. Yes.

📖 How to Use This Calculator

Steps

Enter a number to check: type any integer into the input field. Both positive and negative integers are accepted.

Click Check: the calculator tells you whether the number is a perfect cube, shows its cube root, and displays the nearest perfect cubes above and below.

Switch to List mode for a full sequence: select the List mode and enter an upper limit. The calculator generates every perfect cube from 1 up to that limit in a table.

💡 Example Calculations

Example 1: Is 512 a perfect cube?

Checking a positive perfect cube

n = 512. Compute ³√512 = 8 (exact integer).

Verify: 8 × 8 × 8 = 512. Confirmed.

Previous perfect cube: 7³ = 343. Next perfect cube: 9³ = 729.

512 is a perfect cube. Cube root = 8.

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Example 2: Is 100 a perfect cube?

Checking a non-perfect cube

n = 100. Compute ³√100 ≈ 4.6416 (not an integer).

4³ = 64 and 5³ = 125, so 100 lies between two consecutive perfect cubes.

Previous perfect cube: 64. Next perfect cube: 125.

100 is not a perfect cube. Cube root ≈ 4.6416.

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Example 3: Is −27 a perfect cube?

Checking a negative perfect cube

n = −27. Cube roots of negatives are real and negative.

(−3) × (−3) × (−3) = −27. So ³√(−27) = −3.

−27 is a perfect cube. Cube root = −3.

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Example 4: List all perfect cubes up to 500

Generating the full list of perfect cubes from 1 to 500

Switch to List mode. Enter upper limit = 500.

Floor(³√500) = floor(7.937) = 7. So there are 7 perfect cubes from 1 to 500.

They are: 1, 8, 27, 64, 125, 216, 343 (with cube roots 1 through 7).

7 perfect cubes from 1 to 500: 1, 8, 27, 64, 125, 216, 343.

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❓ Frequently Asked Questions

What is a perfect cube?+

A perfect cube is an integer equal to some integer raised to the third power. For example, 27 is a perfect cube because 3 to the third power equals 27. In other words, n is a perfect cube if and only if the cube root of n is also an integer. The sequence begins 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 for positive perfect cubes.

How do you check if a number is a perfect cube?+

Compute the cube root of the number. If the result is an exact integer, the number is a perfect cube. A reliable method: round the computed cube root to the nearest integer k, then check whether k cubed equals the original number. Rounding avoids floating-point precision issues that could make 7.9999999 appear distinct from 8.

What are the perfect cubes from 1 to 1000?+

The ten perfect cubes from 1 to 1000 are: 1 (1 cubed), 8 (2 cubed), 27 (3 cubed), 64 (4 cubed), 125 (5 cubed), 216 (6 cubed), 343 (7 cubed), 512 (8 cubed), 729 (9 cubed), and 1000 (10 cubed). There are exactly 10 of them because the cube root of 1000 is exactly 10.

Can negative numbers be perfect cubes?+

Yes. Any negative integer whose absolute value is a perfect cube is itself a perfect cube. For example, -8 is a perfect cube because (-2) cubed equals -8. The sequence of negative perfect cubes runs -1, -8, -27, -64, -125, and so on. This is different from perfect squares, which are always non-negative. The cube root function is defined for all real numbers, including negatives.

How many perfect cubes are there from 1 to N?+

There are exactly the floor of the cube root of N perfect cubes from 1 to N. For N = 1000, the floor of the cube root of 1000 equals 10. For N = 100, the floor of the cube root of 100 equals 4, so there are 4 perfect cubes (1, 8, 27, 64). Perfect cubes thin out quickly: up to a million there are only 100, compared to 1000 perfect squares.

What is the difference between a perfect square and a perfect cube?+

A perfect square is a number whose square root is an integer (1, 4, 9, 16, 25...). A perfect cube is a number whose cube root is an integer (1, 8, 27, 64, 125...). Perfect squares are more numerous than perfect cubes: up to N there are about the square root of N perfect squares but only the cube root of N perfect cubes. Numbers that are both are called perfect sixth powers: 1, 64, 729, 4096.

What is a perfect sixth power and how does it relate to perfect cubes?+

A perfect sixth power is a number that is both a perfect square and a perfect cube. It has the form k to the sixth power for some integer k. The first few are 1, 64, 729, 4096, 15625. These arise where the sets of perfect squares and perfect cubes overlap. For example, 64 equals 8 squared (perfect square) and 4 cubed (perfect cube), and also 2 to the sixth (perfect sixth power).

How do perfect cubes relate to the sum of consecutive odd numbers?+

Every perfect cube n cubed equals the sum of n consecutive odd numbers starting at n squared minus n plus 1. For example, 3 cubed equals 27, and the three consecutive odd numbers starting at 7 are 7, 9, 11, which sum to 27. More generally, the nth cube equals the sum of n consecutive odds centred at n squared. This pattern connects cubic numbers to arithmetic sequences and is useful in number theory proofs.

Are 0 and 1 perfect cubes?+

Yes. 0 is a perfect cube because 0 cubed equals 0 (and the cube root of 0 is 0). 1 is a perfect cube because 1 cubed equals 1 (and the cube root of 1 is 1). These are the only two non-negative integers that equal their own cube, and both have integer cube roots, satisfying the definition of a perfect cube. In number theory, 0 and 1 are often treated as trivial perfect cubes.

How do perfect cubes appear in the difference of cubes formula?+

The difference of cubes formula is a cubed minus b cubed equals (a minus b) times (a squared plus ab plus b squared). This factorisation is used in algebra to simplify expressions where both terms are perfect cubes. For example, x cubed minus 8 equals (x minus 2)(x squared plus 2x plus 4) because 8 equals 2 cubed. Recognising perfect cubes lets you apply this formula to factor polynomials efficiently.

What is the fastest way to find the next perfect cube after a given number?+

Compute the cube root of the number and take the ceiling (round up to the next integer). Call this k. Then k cubed is the next perfect cube. For example, to find the next perfect cube after 100: the cube root of 100 is approximately 4.64, so the ceiling is 5, and 5 cubed equals 125. This calculator shows both the previous and next perfect cube for any input in Check mode.

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📌 Quick Tips

💡A number is a perfect cube only if its prime factorisation has every prime appearing a multiple of 3 times. For 8 = 2 cubed, the prime 2 appears 3 times.

💡The difference between consecutive perfect cubes grows quickly: 1, 8, 27, 64, 125, 216 have gaps of 7, 19, 37, 61, 91. There are fewer perfect cubes than perfect squares in any range.

💡Negative integers can be perfect cubes too. -8 = (-2) cubed, -27 = (-3) cubed. The check mode handles negative inputs.

💡To find the nearest perfect cube to any number, round the cube root to the nearest integer and then cube it.

💡In the list mode, setting a limit of 1000 gives all 10 perfect cubes from 1 to 1000 (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000).