Page 31 - Ramanujan Yatra

Ramanujan
YATRA
22
from tuberculosis, his body did not respond to any of the treatment. Moreover, the British climate was not conducive. Added to the misery, the onset of the First World War, meant rationing as well as stoppage of postal parcels with foodstuff from Madras that sustained Ramanujan.
Just a few months earlier, he had been elected member of The Royal Society, given a fellowship in Trinity College, and had made a suicide attempt by jumping before a London Tube. Ramanujan was at the same time elated and despondent.
Godfrey Harold Hardy, the English mathematician, was Ramanujan’s mentor at Cambridge. Although Hardy was shy and cold person, he had a special place for Ramanujan. Wanting to cheer up the dejected Ramanujan ailing in a clinic near London, Hardy wanted to strike a conversion. Later Hardy recollected that incident. “I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxicab No. 1729 and remarked that the number seemed to be rather a dull one and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a fascinating number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.”
13 + 123 = 1 + 1,728 = 1,729
93 + 103 = 729 + 1,000 = 1,729
Amazed at the answer, Hardy asked if he knew the corresponding result for fourth powers: The smallest integer that can be written as a sum of two fourth powers of integers in two ways. After thinking for a moment, Ramanujan replied that he did not know the answer supposed that the first number is huge. Indeed it is a very large number: 1334+1344 = 1584+594= 635,318,657.
This incident launched the “Hardy-Ramanujan number,” or ”taxicab number” into the world of math. Taxicab numbers are the smallest integers which are the sum of cubes in n different ways. The first taxicab number is simple 2 = 13+13. The second is 1729, which can be written as sum of two cubes in two different ways. The third taxi cab number is 87539319, the smallest number that is equal to the sum of two cubes in three different ways. To date, only six taxicab numbers have been discovered.
ENIGMA
Often tales are told to give an impression that Ramanujan used to come up with unusual properties of number by sheer intuition. Often his work is presented as a curiosity, an inconsequential piece of recreational mathematics. Hours of hard work and perseverance lies behind Ramanujanʼs puzzling contributions.