Page 33 - Ramanujan Yatra

Ramanujan
YATRA
24
Ramanujan. Subsequently, Hardy passed on the manuscripts, letters, and notes to G.N. Watson, an eminent mathematician, who came out with 40 papers in the 1920s and 30s. After the death of Watson, the manuscripts remained lodged in Wren library, all but lost, languishing unread for more than 50 years.
While examining the estate of the late G.N. Watson, G.E. Andrews in 1976 accidentally found a box full of manuscripts in Ramanujanʼs distinctive handwriting. It contained about hundred pages, with 138 sides and over six hundred mathematical formulae without proofs. Although technically it was not a notebook, this is the now called the ʻlast notebookʼ. Once rediscovered it opened a flood of new ideas, most notable being mock theta functions, building upon the ideas Ramanujan had recorded in these pages.
More than what meets the eye
Ono and Andrew were amused at first to see the 1+123=93+103 scribbled in a corner. Their immediate reaction could have been just curiosity on this oddity. On careful reading of the sheet, Ono found a set of equations. These equations were familiar; they had the likeness of Fermat’s last theorem. Ramanujan had been working on this grand seventeenth-century problem. “The page mentioned 1729 along with some notes about it. Andrew and I realised that he had found infinitely near misses for Fermatʼs Last Theorem for exponent 3. We were shocked by that,ˮ says Ono.
Pierre de Fermat was one of the most renowned number theorists ever but was too lazy to publish. We come to know much about his work from the letters and notes he had left. Like Ramanujan, he was cryptic and hardly showed the proof. Mathematicians proved one after another the claims of Fermat, except one. Called Fermat's last theorem, it stated that xn + yn = z n has no non-zero integer solutions for x, y and z when n > 2. This remained a riddle for hundreds of years. In the margins of his copy of Diophantus's Arithmetica, Fermat in 1637 had confidently scribbled, ʻI have discovered a truly remarkable proof which this margin is too small to contain’. It took numerous mathematicians, several years of research before the British mathematician Andrew Wiles came up with a proof in 1995 which was 129 pages long!
ELLIPTIC CURVES
If only one can find one x, y, z such that x3+y3=z3; the counter example will prove Fermat wrong. We never know if Ramanujan was trying to prove Fermat wrong, but it is evident that he was computing near misses, that is solutions that are off only by plus or minus one. By finding integer solutions for formulas x3+y3=z3+1 or = z3-1 he dwelled deeper into what is called as ‘elliptic curves’. One of the near misses, Ramanujan offered, was the familiar 1+123=93+103.