Taxicab Numbers or Hardy-Ramanujan Number
We know
about the famous Indian mathematician Srinivasa Ramanujan. On 2nd
May 1918, he was elected as a Fellow of Royal Society. On 13th October
1918, he was the first Indian to be elected as a Fellow of Trinity College,
Cambridge.
Once while
Srinivasa Ramanujan was working with G. H. Hardy at the University of
Cambridge, Hardy visited him in the hospital where Ramanujan was ill. Hardy
mentioned that he had arrived in a taxi numbered 1729, commenting that
it seemed to be “a rather dull number,” and hoped it was not an unlucky omen.
Ramanujan immediately responded, “No, Hardy, it is a very interesting number
indeed. It is the smallest number that can be expressed as the sum of two cubes
in two different ways.”
1729 = 13 + 123
OR
1729 = 93 + 103
Because of
this famous story, 1729 came to be known as the Hardy–Ramanujan
Number.
Numbers that can be written as the sum of two cubes in two distinct ways
are now called taxicab numbers.
The next two
taxicab numbers after 1729 are 4104 and 13832. Try to find the
two different ways in which each of these can be expressed as the sum of two
positive cubes.
4104 = 23
+ 163
OR
4104 = 93
+ 153
And, 13832 =
23 + 243
OR
13832 = 183
+ 203
After these
two numbers, two more taxicab numbers are 20683 and 32832.
These
numbers can be written as the sum of two cubes in two different ways as
follows:
20683 = 103
+ 273 = 193 + 243
32832 = 43
+ 323 = 183 + 303
The number
of taxicab numbers are infinite. You can find many taxicab numbers after these
numbers.
How did
Ramanujan know this? The answer lies in his deep love for numbers. Throughout
his life, he was endlessly curious and constantly played with numerical
patterns. During his years at Cambridge, his colleagues were often astonished
by his remarkable ability to recognize intricate relationships in numbers that
appeared ordinary to others. His friend and collaborator John Littlewood
once said, “Every positive integer was one of Ramanujan’s personal friends.”
Read more
about Srinivasa
Ramanujan.