S. Sivasankaranarayana Pillai, commonly known as S. S. Pillai, was an Indian mathematician who made important contributions to number theory, especially to Waring's problem and related areas of analytical number theory.
Statement by S. S. Pillai at the 36th Annual Session of the Indian Science Congress, 1949.
1. Life History of S. S. Pillai
S. Sivasankaranarayana Pillai was born on April 5, 1901, at Vallam near Courtallam in the Tirunelveli district of Tamil Nadu. His parents were Subbayya Pillai and Gomati Ammal.
His mother died a year after his birth, and he grew up under the care of an old relative in the family. When Pillai was five years old, a teacher was arranged to teach him at home.
At the age of nine, Pillai joined a middle school at Shencottah. There, a teacher called Sastriar recognized his intellectual ability and encouraged him in his studies.
He later moved to a local high school for his matriculation course. During this period, he suffered the sudden loss of his father and faced financial difficulties in continuing his education.
Fortunately, his former teacher Sastriar provided financial assistance. Pillai went to Nagercoil for his Intermediate course at Scott Christian College with a scholarship. He then studied for a B.A. at Maharaja's College in Trivandrum.
In 1927, S. S. Pillai received a research studentship at the University of Madras, where he worked under Professors K. Ananda Rao and R. Vaidyanathaswamy.
He later joined Annamalai University as a lecturer. Much of his important mathematical work was carried out there. The University of Madras awarded him a D.Sc. for his research achievements. He was the first mathematician to receive this distinction from the university.
In 1941, he moved to the University of Travancore. A year later, he joined Calcutta University as a lecturer.
Because of his achievements, Pillai was invited to visit the Institute for Advanced Study at Princeton in the United States for one year. He was also invited to participate in the International Congress of Mathematicians at Harvard University as a delegate of Madras University.
He travelled to the United States in August 1950. Tragically, he died in an air crash near Cairo on August 31, 1950. The Indian mathematical community lost one of its best-known mathematicians, but his mathematical work continues to be remembered.
2. Some Pearls from His Mathematical History
A significant part of S. S. Pillai's mathematical work was completed at Annamalai University. He worked mainly in analytical number theory and made important contributions to Waring's problem.
Waring's problem asks for the smallest number of $k$-th powers of integers needed to represent every positive integer.
In other words, for a given positive integer $k$, one asks for the least number of $k$-th powers required so that every natural number can be written as a sum of such powers.
For example, when $k = 2$, the problem concerns sums of squares. Lagrange proved in 1770 that every positive integer can be represented as the sum of at most four squares.
Hilbert proved in 1909 that the relevant number exists and is finite for every positive integer $k$.
In 1935, S. S. Pillai made an important contribution to Waring's problem by determining the relevant value for a major class of exponents. He also computed precise values in important special cases.
Later developments completed the remaining cases through the work of several mathematicians, including J. Chen and the team of R. Balasubramanian, M. Deshouillers, and F. Dress.
S. S. Pillai also worked on Diophantine approximation and established results concerning the approximation of algebraic and exponential quantities.
His work in this area showed the breadth of his mathematical interests and his ability to connect different parts of number theory.
3. Mathematical Legacy
S. S. Pillai is remembered as one of the pioneers of modern Indian number theory. His work on Waring's problem, analytical number theory, and Diophantine approximation continues to be cited and studied.
His life is also an example of the importance of education, mentorship, and perseverance. Despite difficult personal and financial circumstances, he became one of India's leading mathematicians.
His mathematical contributions remain an important part of the history of number theory in India.