Srinivasa Aaiyangar Ramanujan . He was one of the greatest mathematical geniuses in India. He made substantial contributions to analytical number theory, and worked on elliptic functions, continuous fractions, and infinite series. Member of the Cambridge Philosophical Society.
Summary
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- 1 Biographical synthesis
- 1 Birth
- 2 Studies
- 3 Beginnings as a mathematician
- 4 Theorems and discoveries
- 4.1 Topics where he made notable progress and discoveries
- 5 Death
- 2 The Ramanujan conjecture. Importance
- 3 Ramanujan number
- 4 See also
- 5 Sources
Biographical synthesis
Birth
Srinivasa Ramanujan was born Aaiyangar 22 of December of 1887 in the town of Erode, of the state of Tamil Nadu in India, within a poor and orthodox Brahmin family.
Studies
At age seven he attended a public school on a scholarship. He was a striking self-taught man who recited mathematical formulas and π figures to his classmates. At the age of 12 he was proficient in trigonometry , and at 15 they lent him a book with 6,000 known theorems, without proofs. That was his basic mathematical training. In 1903 and 1907 he suspended the university exams because he was only dedicated to his mathematical diversions.
Beginnings as a mathematician
In 1912 he was encouraged to communicate his results to three distinguished mathematicians. Two of them did not reply, but Godfrey Harold Hardy of Cambridge did . Hardy was about to drop the letter, but the same night he received it he sat down with his friend John Edensor Littlewood (v.) To decipher Ramanujan’s list of 120 formulas and theorems. Hours later they believed they were before the work of a genius. Hardy had his own rating scale for the mathematical genius: 100 for Ramanujan, 80 for David Hilbert , 30 for Littlewood, and 25 for himself.
Some of Ramanujan’s formulas overwhelmed him, but he wrote
” … it is necessary that they be true, because if they were not, no one would have had the imagination to invent them ”
Godfrey Harold Hardy
.
Invited by Hardy, Ramanujan left for England in 1914 and they began working together. In 1917 Ramanujan was admitted to the Royal Society of London and Trinity College, being the first Indian to achieve such an honor.
The main of Ramanujan’s works is in his notebooks, written by him in nomenclature and particular notation, with no proof, which has caused a herculean task of deciphering and reconstruction, not yet completed. Fascinated by the number π, he developed powerful algorithms to calculate it.
Ramanujan worked mainly on the analytical theory of numbers and became famous for his numerous addition formulas related to constants such as π and the natural base of logarithms, prime numbers, and the fraction function of an integer obtained together with Godfrey Harold Hardy .
Theorems and discoveries
Some findings of Ramanujan, and the results obtained in collaboration with Hardy at the beginning of the 20th century :
- Property of highly composed numbers
- Partition functions and their asymptotics
- Ramanujan theta function
Topics where he made remarkable progress and discoveries
- Gamma functions
- Modular forms
- Divergent series
- Hypergeometric series
- Prime number theory
Death
Affected by tuberculosis that was aggravated by the climate of England , Rāmānujan returned to his native country in 1919 and died shortly thereafter in Kumbakonam (260 km from Chennai Madras) at the age of 32. He left several books called Ramanujan Notebooks which are still under study.
Recently, Ramanujan’s formulas have been central to new studies in crystallography and string theory. The Ramanujan Journal is an international publication that publishes works from areas of mathematics influenced by this Indian researcher.
The Ramanujan conjecture. Importance
Although there are numerous expressions that are called the “Ramanujan conjecture”, there is a particularly influential one about the successive works. This Ramanujan conjecture is an assertion regarding the dimensions of the coefficients of the Tau function, a typical peak form in the theory of modular forms. And it has finally been proved subsequently as a consequence of proving Weil’s conjecture by a complicated procedure.
Ramanujan number
Hardy-Ramanujan number is called any natural integer that can be expressed as the sum of two cubes in two different ways. Hardy comments the following anecdote:
-I remember that I went to see him once, when he was already very ill, in Putney. I had taken a taxi bearing the number 1729 and pointed out that such a number seemed uninteresting to me, and I hoped that he would only make a dismissive sign
“No” – he answered me – this is a very interesting number; is the smallest number that we can decompose in two different ways with the sum of two cubes. GH Hardy
Indeed, 93 + 103 = 13 + 123 = 1729
Other numbers possessing this property had been discovered by the French mathematician Bernard Frénicle de Bessy (1602-1675):
- 23 + 163 = 93 + 153 = 4104
- 103 + 273 = 193 + 243 = 20683
- 23 + 343 = 153 + 333 = 39312
- 93 + 343 = 163 + 333 = 40033
The smallest of the numbers decomposable in two different ways in addition of two powers to the fourth is 635 318 657, and it was Euler (1707-1763) who discovered it:
- 1584 + 594 = 1334 + 1344 = 635318657
The smallest number that can be expressed as a sum of two non-zero positive cubes n in two different ways to the order of the operands is called the nth Taxicab number, denoted as Ta (n) or Taxicab (n). Such that Ta (1) = 2 = 13 + 13, Ta (2) = 1729 and Ta (3) = 87539319. Variant of the taxicab is the cabtaxi (a cabtaxi number is defined as the smallest integer that can be written in n different ways (in the order of approximate terms) as the sum of two positive, null or negative cubes).
