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Dept. of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI) has organised an colloquia on “Fermat’s Two Square Theorem, Quadratic Forms, Brahmagupta’s Bhavana and Bhargava’s Cubes” by Professor Vijay Patankar, FLAME university, Pune on 16 October 2025.

Title: Fermat’s Two Square Theorem, Quadratic Forms, Brahmagupta’s Bhavana and Bhargava’s Cubes

Abstract of First Talk:

Fermat’s Two Square Theorem (1640) says that any prime number that leaves a remainder 1 when divided by 4 can be expressed as a sum of two squares of integers. There are many proofs of this fact. Almost each proof carries, in seed form, some deeper concepts or techniques from mathematics. In this first talk, we will prove this theorem using method of reduction of Quadratic forms and using the action of the group of 2 × 2 integer matrices with determinant 1. As an aside, we will try to explain the notion of composition of Quadratic forms, also known as Bh ̄avan ̄a, a Sanskrit word that was used by ancient Indian algebraists to name a principle of “composition” introduced by Brahmagupta (born in 598 CE) in the seventh century. Manjul Bhargava’s work on the Higher composition laws, a far reaching generalization of Bh ̄avan ̄a, earned him the Fields medal in 2014.