Posts Tagged 'Ramanujan'

Discoveries by Amateurs and Distractions by Cranks

Do amateurs ever solve outstanding mathematical problems? Professional mathematicians are aware that almost every new idea they have about a mathematical problem has already occurred to others. Any really new idea must have some feature that explains why no one has thought of it before  [TM155 or search for “thatsmaths” at irishtimes.com].

fermat-ramanujan

Pierre de Fermat and Srinivasa Ramanujan, two brilliant “amateur” mathematicians.

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Ramanujan’s Astonishing Knowledge of 1729

Question: What is the connection between Ramanujan’s number 1729 and Fermat’s Last Theorem? For the answer, read on.

The story of how Srinivasa Ramanujan responded to G. H. Hardy’s comment on the number of a taxi is familiar to all mathematicians. With the recent appearance of the film The Man who Knew Infinity, this curious incident is now more widely known.

K3-Surfaces-Google

Result of a Google image search for “K3 Surface”.

Visiting Ramanujan in hospital, Hardy remarked that the number of the taxi he had taken was 1729, which he thought to be rather dull. Ramanujan replied “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

Continue reading ‘Ramanujan’s Astonishing Knowledge of 1729′

Waring’s Problem & Lagrange’s Four-Square Theorem

\displaystyle \mathrm{num}\ = \square+\square+\square+\square

Introduction

We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved numerous ingenious and beautiful results in partition theory.

More generally, additive number theory is concerned with the properties and behaviour of integers under addition. In particular, it considers the expression of numbers as sums of components of a particular form, such as powers. Waring’s Problem comes under this heading.

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Ramanujan’s Lost Notebook

In the Irish Times column this week ( TM010 ), we tell how a collection of papers of Srinivasa Ramanujan turned up in the Wren Library in Cambridge and set the mathematical world ablaze. Continue reading ‘Ramanujan’s Lost Notebook’


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