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An outstanding mathematician

SUBBAYYA SIVASANKARANARAYANA Pillai (1901- 1950), a brilliant mathematician known for his remarkable work in Number Theory, was born on April 5,1901, at Vallam near Courtallam town in Tamilnadu. His mother Gomathi Ammal died within a year after his birth and he grew up under the care of an old relative. When Pillai reached the age of five, his father Subbayya Pillai, made arrangements for his education at home for the next three years. From around the age of nine, Pillai was put in the middle school at nearby Shencottah.

During his matriculation he received a shattering blow from his father's sudden death. He lacked the wherewithal even to continue his high school education. Fortunately, his former teacher Sastriar came to his rescue with monetary support to complete his school education and also go beyond. Pillai secured a scholarship to do his Intermediate course in the Scott Christian College at Nagercoil and later, his B.A. too at Maharaja's College, Trivandrum.

After Pillai shifted to Madras around 1927, a research studentship in the University of Madras materialized. Also, the mathematical guidance from Professors K.Ananda-Rau and R.Vaidyanathaswamy, the imprint of which was evident already in Pillai's early mathematical papers.

A good part of his more than seventy mathematical publications was written by him during his stay as Lecturer in Annamalai University from 1929 to 1941. This includes his famous contribution to the solution of Waring's problem, aptly described in 1950 by Professor K.Chandrasekharan as "almost certainly his best piece of work and one of the very best achievements in Indian Mathematics since Ramanujan".

It was Lagrange who published,in 1770, a proof for Fermat's statement of 1640 that every natural number is a sum of 4 squares of integers (integers being the numbers 0,1,-1,2,-2,3,-3 and so on) . In other words, to express any natural number n as a sum of s squares of integers, one may take s to be 4 independently of whatever n may be. Waring's problem is the generalization of this beautiful result in arithmetic to the case of k-th powers of integers instead of squares (i.e. 2-nd powers) of integers.

In 1770, Waring, Lucasian Professor of Mathematics in Cambridge University, stated in his Meditationes Algebraicae, that every number is the sum of 4 squares, 9 cubes, 19 biquadrates and so on. Let g(k) be the least value of s such that every (natural) number (say, n) is the sum of s k-th powers..... Waring's assertion means not only that g(k) is independent of n ,but,also that g(2)=4, g(3)=9, g(4)=19 and so on. But Waring's result was only a conjecture, and he had no proof of it. The existence of g(k),for every k, was proved by D.Hilbert in 1909.

Incidentally, the alert reader can easily verify that g(2), g(3), g(4) cannot be less than 4, 9, 19 respectively, by simply trying to express 7 (or 15) as sum of squares,23(or 239) as a sum of cubes and 79 as sum of biquadrates (i.e. fourth powers) of natural numbers.

For any natural number k bigger than 1, denote the k-th power of 3/2 by m,say and the largest natural number not exceeding m , say,by l. For example, if k =3, then m=(27)/8, l=3 and if k=4,we have m= (81)/16, l=5. In 1772, Euler ( son of the famous Leonhard Euler) showed that g(k) cannotbe less than the sum L of l-2 and the k-th power of 2 ,i.e. g(k) can only equal L or exceed it at best. Pillai obtained, independently of L.E.Dickson, an almost complete solution of Waring's problem in 1935, viz. that g(k) equals L ,for all values of k larger than 6,under the assumption of the validity of a highly plausible inequality involving m-l ( the 'fractional part' of m, the k-th power of 3/2).

Pillai also proved that g(6)= 73, the precise value of L above for k=6 ,while Jing-run Chen showed in 1965 that g(5)= 37 ( also the value of L for k=5), as conjectured. Pillai's papers on Waring's problem reveal his uncanny ability to triumph with virtually the thrill or excitement like in a photo-finish win at a competition. It is not out of place now to briefly describe the more recent confirmation of the value of g(4) as 19. In 1928, L.E.Dickson showed that that g(4) equals at most 30, which was improved to 22 by H.E.Thomas in 1973-'74 and further to 20 by R.Balasubramanian. In 1985, along with M.Deshouillers and F.Dress,Balasubramanian created a stir by capturing the expected value 19 for g(4).

On the basis of a result of his on Diophantine approximation, Pillai formulated (a still open) conjecture on the finiteness of the number of integer solutions for an `exponential' Diophantine equation which subsumes the Ramanujan-Nagell equation and Catalan's equation. Recently, it was shown by T.N.Shorey that the "Generalized a,b,c Conjecture" implies Pillai's (aforementioned) conjecture. The former-yet to be confirmed- is a formidable conjecture also implying the truth of a weaker version ofthe venerable Fermat's last theorem. Pillai had extended some of Ramanujan's work on highly composite numbers, already in 1933 but got it published in 1944,only after similar results due to ErdOs and Alaoglu came to be known.

Number Theory was, for Pillai, an inexhaustible source of problems but Pillai was also seriously interested in tackling famous tough problems in other areas. One could discern striking originality and innovative approach in his papers which rarely needed to cite from elsewhere. Pillai's mathematical achievements seem to have peaked during his stay with Annamalai University and it was also then that he got his D.Sc degree from Madras University the first such for Mathematics there from.

In August 1950, he proceeded by air to the US., to be a visiting member for a year at the Institute for Advanced Study, Princeton, and also as a delegate of Madras University, to participate in the International Congress of Mathematicians at Harvard University. But due to the air crash near Cairo he was lost forever. Sadly, he failed to get the recognition that he richly deserved from any of our Science Academies.

S.Raghavan

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