Page 32 - Ramanujan Yatra

It is not such a big surprise that Ramanujan knew that 1729 is sum of two cubes that can be expressed in two different ways. Since Ramanujan was from colonial India, which used Imperial measures at the time, he would probably have been taught at school that there were 1728 cubic inches in a cubic foot. He would also have known that 729=93. But how he knew that it was the smallest?
Properties of 1729 were not unknown to Ramanujan. He actually had written it down before even coming to England. In his second notebook, in chapter XVIII, the case is given as a single worked-out example of an infinite family of solutions for the Diophantine equation of the form x3+y3=u2 and Euler's equation x3+y3+z3=u3. Ramanujan provides six numerical illustrations for the first and 12 for the second. The second of the 12 examples for Euler's equation is the famous 13+123=93+103 (=1729).
This also finds a place in questions and solutions submitted by Ramanujan to the Journal of the Indian Mathematical Society. Once in 1913 as a solution to the identity (6A2−4AB+4B2)3=(3A2+5AB−5B2)3+(4A2−4AB+6B2)3 +(5A2−5AB−3B2)3. Dividing both sides by 27, Ramanujan arrives at 123 = (−1)3 + 103 + 93. Again in 1915 (question no 681, vol VII page 160), he seeks the solution in integers for the equation x3+y3+z3=1. He asks the readers to deduce 93+103=123+1 from the general solution to the identity.
What Ono had stumbled upon was the fourth entry of this now-famous number, until then unnoticed.
Lost and found ʻthe last notebook’
Given his health, mental condition and the grim situation resulting from the breaking of the First World War, Hardy decided that returning to India is the best option for Ramanujan. Hardy made arrangements and Ramanujan returned to India emaciated. Since his return, until his death on April 22, 1920, his wife Janaki took care of him.
Even while recouping, Ramanujan continued his mathematical explorations. Janaki recalled how Ramanujan used to claim that mathematics was his distraction from the pain and sufferings. He wrote theorems and formulas like there is no other day. After Ramanujan died, the young widow collected the three notebooks and sheaf of papers and submitted them to the University of Madras. The University made a copy of these papers and a set to Hardy who was then embarked upon producing a posthumous volume on Ramanujan containing published documents, notebooks, and other unpublished work. Hardy, in association with P. V. Seshu Aiyar, came out with Collected Papers of Srinivasa
Ramanujan
YATRA
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