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Dept. of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI) has organised an colloquia on “Fermat’s Two Squares Theorem and modular forms” by Professor Vijay Patankar, FLAME university, Pune on 17 October 2025.

Title: Fermat’s Two Squares Theorem and modular forms

Abstract for Second Talk:

In this talk, we will explore another approach to proving the Fermat’s Two Square theorem. We will first consider a very natural generating function which we will then view as a complex function. We will then show that this new function satisfies a special symmetry property thanks to a theorem from Fourier Analysis called the Poisson Summation Formula making it a modular form. In fact, we will show that it belongs to a specific vector space of modular forms of weight 1 and level 4. This vector space happens to be of dimension one and has a natural basis. The resulting equality of vectors proves Fermat’s Two Square theorem! Walk further and the same technique then leads us to Lagrange’s Four Square Theorem (1770): Every positive integer is a sum of four squares of integers!