The taxicab number
Two different ways to make a sum of two cubes.
A dull number?
Hardy rode in a taxi numbered 1729 and called it “rather a dull number”. Ramanujan said it was very interesting. It is a sum of two cubes.
In how many different ways can 1729 be written as a sum of two cubes?
What this lesson covers
The idea
A number that can be expressed as the sum of two cubes in two different ways is a taxicab number; the smallest is 1729 = 1³ + 12³ = 9³ + 10³.
A dull number?
Hardy rode in a taxi numbered 1729 and called it “rather a dull number”. Ramanujan said it was very interesting. It is a sum of two cubes.
In how many different ways can 1729 be written as a sum of two cubes?
- In no way
- In exactly one way
- In two different ways
Split 1729 into cubes
Tap a cube. The toy takes it away from 1729 and checks whether what is left is a cube too.
Two ways
A number that can be written as the sum of two cubes in two different ways is a taxicab number. The smallest is 1729 = 1³ + 12³ = 9³ + 10³.
Because of this story, 1729 is called the Hardy–Ramanujan number. To hunt for a way, take a cube away from the number and check whether the rest is a cube.
- Way | Cubes | Sum
- 1³ + 12³ | 1 + 1728 | 1729
- 9³ + 10³ | 729 + 1000 | 1729
Notes
A number that can be written as the sum of two cubes in two different ways is a taxicab number; the smallest is 1729 = 1³ + 12³ = 9³ + 10³.
Check yourself
Which is the right way to write 1729 as a sum of two cubes?
Work out 9^3 + 15^3.
Answer: 4104
9³ = 729 and 15³ = 3375, so the sum is 4104. Also 2³ + 16³ = 8 + 4096 = 4104, so 4104 is a taxicab number too.
35 = 2³ + 3³, and 35 − 1³ = 34 is not a cube, so this is its only way. Is 35 a taxicab number?
Why is 1729 special?
- 1 + 12. 1 + 12 = 13 adds the numbers. We add their cubes: 1³ + 12³.
- 1³ + 12³ — correct. Yes! 1 + 1728 = 1729.
- (1 + 12)³. (1 + 12)³ = 13³ = 2197. Cube each number first, then add.
- 1² + 12². That adds squares: 1 + 144 = 145. A cube multiplies a number three times.
- Yes, it is a sum of two cubes. One way is not enough. A taxicab number needs two different ways.
- No, it has only one way — correct. Yes! A taxicab number needs two different ways, and 35 has only 2³ + 3³.
- Yes, because 2 and 3 are different. It is not about the two numbers being different. We need two different pairs of cubes with the same sum.
- It is a perfect cube. 1729 is just 1 more than 12³ = 1728, so it is not a cube.
- It is the smallest number that is a sum of two cubes in two different ways — correct. Yes! 1729 = 1³ + 12³ = 9³ + 10³, and no smaller number does this.
- It is a sum of two cubes in only one way. It has two ways: 1³ + 12³ and 9³ + 10³.