Joomla! site syndication http://mathematics.science-tips.org 2008-08-29T02:58:27+01:00 http://mathematics.science-tips.org http://mathematics.science-tips.org/images/M_images/joomla_rss.png text/html 2006-11-02T03:08:04+01:00 http://mathematics.science-tips.org http://mathematics.science-tips.org/news/latest-news/indian-professor-suggests-solutions-to-fermats-last-theorem.html The News: Prof V.K. Gurtu (66), a retired pro¬fessor and former head of the mathematics department at Laxminarayan Institute of Tech¬nology, Nagpur, India has claimed to have given two solu¬tions to Fermat’s Last Theorem (FLT). The 400 year old problem, which has its origin in the Historic Marginal Note (HMN), by Fermat (1601-1665) puzzled mathe¬maticians world over since then. In the 28-page paper titled “On Fer¬mat’s Historic Marginal Note: Some Left Out Grains of Truth Leading to New Proof of FLT”, Prof. Gurtu have offered two solutions. One of them is based on the 17th century techniques, in the times of Pierre de Fermat time while, another is beased on modem methods. This paper is presented at the prestigious International Congress of Mathematicians, Madrid in August. The Background: Pierre de Fermat is a French judge and mathematician. In problem II.8 of his Arithmetica, Diophantus, a mathematician, asks how to split a given square number into two other squares. In other words, given a rational number k, find u and v, both rational, such that k2 = u2 + v2. He also shows how to solve this for k = 4. In 1637 Fermat wrote a comment (in Latin) in the margin of this problem in his copy of the Arithmetica (This 1621 edition, translated from Greek into Latin by Claude Gaspard Bachet de Méziriac). The original comment by Feramt read: “Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.” It translates to “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” This later came to be known as the “Historic Marginal Note (HMN) by Fermat.” The proposition, exponded there became famous as the “Fermat’s Last Theorom.” Here, you may like to know that the theorem was not the last, to be conjectured by Fermat, but the last to be proved. Fermat's Last Theorem became one of the most famous theorems in the history of mathematics. It states that: “It is impossible to separate any power higher than the second into two like powers” or mathematically, “If an integer n is greater than 2, then an + bn = cn has no solutions in non-zero integers a, b, and c.” It is strikingly similar to the widely proved Pythagoras Theorem but its simplicity also gave many incorrect proofs. No correct proof was found for 357 years, until it was finally proven using very sophisticated methods by Andrew Wiles in 1995. The Present Scene: Now, Prof. Gurtu claims to have revealed facts which were left out by earliest researchers including Euler, Gauss, Dirichlet and Legendre who have proved the FLT to the fifth power of any number. Press Trust of India, PTI reports that in the first proof, Prof. Gurtu used identities known in the time of Fermat and the famous Fermat’s method of infinite descent, while in the sec¬ond, non-natural numbers said to have been used for a very limited pur¬pose. It is interesting to note that Prof. Gurtu claims that his tech¬niques are closest to Fermat’s thought process and it is under examination at an American peer review journal. “The non-natural numbers eliminate themselves in the course of the solution of the equation without any difficulty,” he also said and added that, “he had been working on the problem since 1989, offered a solution in 1998 to the Indian Mathematical Society and to whose related queries, found satisfactory answers in the past eight years.”