Taxicab number facts for kids

A taxicab number is a special kind of number that mathematicians find fascinating. It's the smallest number that can be made by adding up two positive cubes in a certain number of different ways. The name "taxicab number" comes from a famous story about two brilliant mathematicians, Godfrey Hardy and Srinivasa Ramanujan.

The Story of Hardy and Ramanujan's Taxi

Godfrey Hardy was a mathematics professor at Cambridge University. One day, he went to visit his friend, the amazing young Indian mathematician Srinivasa Ramanujan, who was unwell. Both men loved thinking about numbers.

When Ramanujan heard that Hardy had arrived in a taxi, he asked about the taxi's number. Hardy said it was just a "boring" number: 1729. But Ramanujan quickly replied that 1729 was not boring at all! He said it was a very interesting number. He explained that it was the smallest number that could be made by adding two cubes together in two different ways.

This story is very well-known among mathematicians. Because of this famous conversation, 1729 is sometimes called the "Hardy-Ramanujan number."

What Makes 1729 Special?

To understand taxicab numbers, you need to know about "cubes."

• When a number is multiplied by itself, the answer is called a "square." For example, 3 multiplied by 3 (3x3) equals 9. So, 9 is a square.
• When a number is multiplied by itself three times, the answer is called a "cube." For example, 3 multiplied by 3, then by 3 again (3x3x3) equals 27. So, 27 is a cube.
• Another example of a cube is 8, because it's 2x2x2.
• When we add two cubes together, like 27 + 8 = 35, we call 35 the "sum of two cubes."

The number 1729 is special because there are two different ways to show it as the sum of two cubes:

• First way: 1 multiplied by itself three times (1x1x1) equals 1. And 12 multiplied by itself three times (12x12x12) equals 1728. So, 1 + 1728 = 1729.
• Second way: 9 multiplied by itself three times (9x9x9) equals 729. And 10 multiplied by itself three times (10x10x10) equals 1000. So, 729 + 1000 = 1729.

Many other numbers can also be shown as the sum of two cubes in more than one way. But 1729 is the very smallest number that can be done in two different ways.

Other Amazing Taxicab Numbers

Since the famous conversation between Hardy and Ramanujan, mathematicians have been looking for other taxicab numbers. They want to find the smallest numbers that can be expressed as the sum of two cubes in three, four, five, or even more different ways!

These numbers get very, very big. Scientists often use powerful computers to help them find these huge numbers.

Here are the first two taxicab numbers:

• Ta(1) = 2. This is 1³ + 1³. (It's the smallest number that can be written as the sum of two cubes in one way).
• Ta(2) = 1729. This is 1³ + 12³ and also 9³ + 10³. (It's the smallest number that can be written as the sum of two cubes in two ways).

Mathematicians have found more taxicab numbers, like Ta(3), Ta(4), Ta(5), and Ta(6). These numbers are incredibly large, with many digits! For example, Ta(6) has 23 digits! Each of these numbers is the smallest number that can be written as the sum of two cubes in that many different ways.

See also

In Spanish: Número taxicab para niños