Fundamental Theorem of Algebra: Two Proofs

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Introduction

Fundamental theorem of algebra is one of the most famous results provided in higher secondary courses of mathematics. Normally it is mentioned in chapter related to complex numbers where the reader is made aware of the power of complex numbers in solving polynomial equations. The theorem guarantees that any non-constant polynomial with real or complex coefficients has a complex root.

Congruence Properties of Partitions: Part 2

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Continuing our journey of partition congruences from the last post we now prove the congruences modulo $7$ and $11$.

Congruence Properties of Partitions: Part 1

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Introduction

We know that any positive integer greater than $1$ can be expressed as a product of prime numbers in a unique fashion ignoring the order of factors. This is one of the most basic results in number theory and is aptly called the fundamental theorem of arithmetic. This result also shows that prime numbers are the building blocks for all integers and this justifies the importance given to prime numbers in number theory.

Thoughts On Ramanujan

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Of late I had been reading Ramanujan's Collected Papers and based on my understanding of it (and inputs from works of Borwein brothers, Bruce C. Berndt) I wrote a series of posts explaining some of Ramanujan's discoveries (see 10 posts starting from here and 4 posts beginning from here). While studying Ramanujan's Papers I could not help myself being astounded by the depth of his discoveries and the ingenuity of the proofs he provided for some of his results.

Reading Papers has not been an easy job for me and seems like an unending task if I wish to have a complete and thorough understanding of it. Hence I decided to take a break for sometime and dedicate one of my posts about my thoughts on Ramanujan, his works, abilities and methods. Needless to say whatever I present here would be a personal view and may differ from general perception a reader might have of Ramanujan and his works. Because of the same reason this post is bound to be of somewhat personal nature.

Proof of Chudnovsky Series for 1/π(PI)

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In 1988 D. V. Chudnovsky and G. V. Chudnovsky (now famous as "Chudnovsky Brothers") established a general series for $\pi$ by extending Ramanujan's ideas (presented in this series of posts). It can be however shown that their general series can be derived using Ramanujan's technique. Chudnovsky's approach has the advantage that using class field theory the algebraic nature of parameters in the general series can be determined and this greatly aids in the empirical evaluation of the parameters and thereby providing an actual series consisting of numbers.