**Date & Time:** Wednesday, January 15, 2014, 16:00-17:00.

**Venue:** Ramanujan Hall

**Speaker:** N. Saradha, TIFR Mumbai

**Title:** Number of representations of integers by binary forms

**Abstract:** Let F(x,y) be an irreducible binary form of degree r >= 3 with integer coefficients and h a non-zero integer. In a seminal work in 1909, Thue proved that the equation
F(x,y) = h
has only fnitely many solutions in integers x and y. For this purpose, he employed a method based on approximation of algebraic numbers by rationals. His method was developed by several mathematicians to give better estimates for the number of solutions of Thue equations. In 1987, Bombieri and Schmidt estimated the number of primitive solutions as
cr^{1+w(h)},
where c is an absolute constant and w(h) denotes the number of distinct prime divisors of h. Further they showed that c = 215 if r >= r_{0} where r_{0} is unspecified. In a joint work with Divyum Sharma, we showed that the above result of Bombieri and Schmidt is true with r_{0} = 23. Further, c may be taken as small as 10 provided the discriminant of F is large compared to r. In this talk, I shall indicate some salient features of Thue's method and how the improvement in our work has been obtained.