1729

2023-06-30

G. H. Hardy was an influential English mathematician in the 20th century. Hardy's greatest contribution to mathematics, as stated by himself, was his experience as the mentor of the brilliant Indian mathematician, Srinivasa Ramanujan.

Without going into a lot of detail about Ramanujan's life, Ramanujan was a child prodigy in mathematics and moved from India to Cambridge, England to work with Hardy in 1914.

Ramanujan had many health problems throughout his life, and his condition worsened during his time in England. One day when Ramanujan was sick, Hardy took a taxi to go visit him. Hardy noted that he had ridden in taxicab number 1729. He then remarked to Ramanujan that 1729 seemed like a dull number. Most people would have likely agreed with this opinion, as 1729 does not describe a prime number, nor a highly composite number. It isn't a Fibonacci number. It isn't a power of 2. It isn't even an even number. Hardy hoped that the dullness of the number 1729 wasn't a bad omen. Ramanujan then corrected Hardy, mentioning that "No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

Because of this brilliant interaction, 1729 is known best as Ramanujan's number or the Ramanujan-Hardy number. It also gave rise to a new set of numbers. The Taxicab numbers: the nth Taxicab number being the smallest positive integer that can be expressed as the sum of 2 positive cubes in n distinct ways. Note the condition of positivity. Ramanujan was arguably flawed in his statement, as he should have specified that the cubes must be positive. Otherwise, 91 would be the correct number.