arXiv subjects
Publications and source records attributed to Manjil P. Saikia.
In this note, we shall give a brief survey of the results that are found in Ramanujan's Lost Notebook related to cranks. Recent work by B. C. Berndt, H. H. Chan, S. H. Chan and W. -C. Liaw have shown conclusively that cranks was the last mathematical object that Ramanujan studied. We shall closely follow the work of Berndt, Chan, Chan and Liaw and give a brief description of their work.
In this paper we propose a definition of a recurrence relation homomorphism and illustrate our definition with a few examples. We then define the period of a k-th order of linear recurrence relation and deduce certain preliminary results associated with them.
We give a brief review of the work of Ramanujan on cranks that is found in the Lost Notebook. Recent work by Bruce Berndt and his coauthors have brought to light many interesting results of Ramanujan on cranks, which we highlight in this article.
This is the first of the proposed sets of notes to be published in the website Gonit Sora (http://gonitsora.com). The notes will hopefully be able to help the students to learn their subject in an easy and comprehensible way. These notes are aimed at mimicking exactly what would be typically taught in a one-semester course at a college or university. The level of the notes would be roughly at the undergraduate level. The present sets of notes are not yet complete and this is the second version that is being posted. These notes contain very few proofs and only state the important results in Probability Theory. These notes are based on the course taught at Tezpur University, Assam, India by Dr. Santanu Dutta. There may be some errors and typos in these notes which we hope the reader would bring to our notice.
We give a brief historical overview of the famous Pythagoras' theorem and Pythagoras. We present a simple proof of the result and dicsuss some extensions. We follow \cite{thales}, \cite{wiki} and \cite{wiki2} for the historical comments and sources.
We study Kummer's approach towards proving the Fermat's last Theorem for regular primes. Some basic algebraic prerequisites are also discussed in this report, and also a brief history of the problem is mentioned. We review among other things the Class number formula, and use this formula to conclude our study.
We give a new proof of Lucas' Theorem in elementary number theory.
We try to improve a problem asked in an Indian Math Olympiad. We give a brief overview of the work done in Saikia and Vogrinc (2011) and Saikia and Vogrinc where the authors have found a periodic sequence and the length of its period, all inspired from an Olympiad problem. The main goal of the paper is to improve the main result in Saikia and Vogrinc (2011).
In this short note we prove a result that is an extension of an old Olympiad problem and is a very simple variant of the question of finding an approximation for $k$, where it is a nonzero constant and it satisfies the equation $a^k+b^k=c$, where $a,b,c$ are all real constants.
We derive an interesting congruence relation motivated by an Indian Olympiad problem. We give three different proofs of the theorem and mention a few interesting related results.