# 1729

Hardy (left) and Ramanujan (right)

Indian mathematician, Srinivasa Ramanujan, was one of the greatest number theorists. He was born into a poor Brahman family, and with no formal education. Although Ramanujan was able to find a few mathematics textbooks, he didn’t have enough material and thus had to find the solutions by himself. In doing so, he independently stated 6165 theorems – some that had already been discovered in the West and some that hadn’t. In 1914, he arrived at Cambridge on a scholarship due to G.H. Hardy’s request to work with him. However, because of the difference in the English culture and weather with that of India, Ramanujan was very unhappy and fell ill with tuberculosis.

One day, when Hardy visited Ramanujan in hospital and told Ramanujan that he had travelled in a black cab with the number 1729, apologising about the fact that is was a  “rather dull one [number]” and hoping it wouldn’t bring an “unfavourable omen”. However, Ramanujan turned to Hardy and exclaimed:

“No, Hardy, it’s a very interesting number! It’s the smallest number expressible as the sum of two cubes in two different ways.”

Due to this story 1729 became famous in the mathematical circle and is called the Hardy–Ramanujan number. This story led to the establishment of ‘taxicab numbers‘, which are the smallest numbers that can be expressed as the sum of two cubes in n distinct ways. Only six other taxi-cab numbers have been found that share the same properties.

Some other interesting facts about 1729:

• It’s a ‘Fermat near miss’: numbers of the form 1 + z3 which are also expressible as the sum of two other cubes
• It’s a Carmichael Number
• It’s the first Euler pseudoprime
• It’s a spherical number: a positive integer that is the product of three distinct prime numbers
•  Masahiko Fujiwara showed that 1729 is one of four positive integers, which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:
1 + 7 + 2 + 9 = 19
19 × 91 = 1729

Further Exploring:

Numberphile Video

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