When the images are uploaded, you can adjust animation speed and frame order before making the GIF. After generating the GIF, you can resize, crop, and optimize it or use any of the other tools we provide. Animated GIF:. How to make a GIF? By default, all frames will be sorted in alphabetical order, you can drag and drop them inside GIF animator area to rearrange. It was finally proven in , as a consequence of Pierre Deligne 's proof of the Weil conjectures.
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The reduction step involved is complicated. Deligne won a Fields Medal in for that work.
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This congruence and others like it that Ramanujan proved inspired Jean-Pierre Serre Fields Medalist to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. Deligne in his Fields Medal-winning work proved Serre's conjecture.
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The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory there would be no proof of Fermat's Last Theorem. While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of looseleaf paper. They were mostly written up without any derivations. This is probably the origin of the misapprehension that Ramanujan was unable to prove his results and simply thought up the final result directly.
Mathematician Bruce C. Berndt , in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, but chose not to.
This may have been for any number of reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate , and then transfer just the results to paper. Using a slate was common for mathematics students in the Madras Presidency at the time. He was also quite likely to have been influenced by the style of G.
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Carr 's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results. The first notebook has pages with 16 somewhat organised chapters and some unorganised material. The second has pages in 21 chapters and unorganised pages, and the third 33 unorganised pages.
The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as did G. Watson , B. Wilson , and Bruce Berndt. The number is known as the Hardy—Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words: . I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number and remarked that the number seemed to me rather a dull one , and that I hoped it was not an unfavorable omen.
Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends. Generalizations of this idea have created the notion of " taxicab numbers ". In his obituary of Ramanujan, written for Nature in , Hardy observed that Ramanujan's work primarily involved fields less known even among other pure mathematicians, concluding:. His insight into formulae was quite amazing, and altogether beyond anything I have met with in any European mathematician. It is perhaps useless to speculate as to his history had he been introduced to modern ideas and methods at sixteen instead of at twenty-six.
It is not extravagant to suppose that he might have become the greatest mathematician of his time. What he actually did is wonderful enough He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account.
Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to Hardy gave himself a score of 25, J. Littlewood 30, David Hilbert 80 and Ramanujan In his book Scientific Edge , the physicist Jayant Narlikar spoke of "Srinivasa Ramanujan, discovered by the Cambridge mathematician Hardy, whose great mathematical findings were beginning to be appreciated from to His achievements were to be fully understood much later, well after his untimely death in For example, his work on the highly composite numbers numbers with a large number of factors started a whole new line of investigations in the theory of such numbers.
The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence. Stamps picturing Ramanujan were issued by the government of India in , , and The International Centre for Theoretical Physics ICTP has created a prize in Ramanujan's name for young mathematicians from developing countries in cooperation with the International Mathematical Union , which nominates members of the prize committee.
House of Ramanujan Mathematics, a museum of Ramanujan's life and work, is also on this campus. In , on the th anniversary of his birth, the Indian government declared that 22 December will be celebrated every year as National Mathematics Day. Situated next to the Tidel Park, it comprises 25 acres with two zones, with a total area of 5. From Wikipedia, the free encyclopedia. This is the latest accepted revision , reviewed on 7 October For other uses, see Ramanujan disambiguation.
Indian mathematician. In this Indian name , the name Srinivasa is a patronymic , not a family name , and the person should be referred to by the given name , Ramanujan. Hardy J.
Main article: Ramanujan—Petersson conjecture. Further information: Ramanujan's lost notebook. Main article: number. Further information: List of things named after Srinivasa Ramanujan. Berndt, Bruce C. Butzer, P. Turnhout, Belgium: Brepols Verlag. Ramanujan's Lost Notebook. Part I. New York: Springer. Part II. Part III. Part IV. Ramanujan: Letters and Commentary. Ramanujan: Essays and Surveys. Number Theory in the Spirit of Ramanujan. Ramanujan's Notebooks.