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June - July 2008


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FERMATíS LAST THEOREM: GURTUíS 17th CENTURY ELEGANT PROOF

by Kishan Sharma


Fermat’s Last Theorem (FLT) has its origin in the Historic Marginal Note (HMN) which the French jurist and mathematician Pierre de Fermat (1601 – 1665) had written around 1637 in the backdrop of Pythagoras Theorem in its arithmetic form. In modern terminology, both these theorems can be jointly stated as follows. “The equation x raised to power n plus y raised to power n equal to z raised to power n is solvable in positive integers for n = 2 but not for its higher values”. The last sentence of this HMN was : “I have a truly marvelous demonstration of this proposition but this margin is too narrow to contain it”. But the demonstration (proof) of this proposition (FLT) couldnot be traced out even after Fermat’s death. Surprisingly, in spite of innumerable efforts made by top mathematicians of the world, Fermat’s original proof of FLT couldnot see the light of the day until Dr. V. K. Gurtu (Retd.) Professor & Head of Mathematics Department of Laxminarayan Institute of Technology (LIT), RTM Nagpur University, Nagpur (India), presented it at the last International Congress of Mathematicians (ICM) held at Madrid in Spain. Later on, Dr. Gurtu incorporated this proof in his recently published book entitled “FLT and some other outstanding number theory problems with their arithmetical solutions”. Dr. Gurtu has dedicated this book published by Himalaya Publishing House, India; to the memory of Fermat and accordingly it was formally released on the eve of his 406th birth anniversary by RTM Nagpur University Vice Chancellor Dr. S. N. Pathan in presence of Pro-VC Dr. G.S. Parasher and Director of LIT Dr. R.B. Mankar.

Dr. Gurtu firmly believes that the real solution of any problem, mathematical or otherwise, can be found only by going into its genesis. Accordingly, for discovering originator’s proof of FLT, he has used those methods, which are used for obtaining Pythagorean triples and Fermat’s own unique method of infinite descent. In short, Dr. Gurtu’s approach can be described as follows. In 1770, Swiss mathematician Leonhard Euler (1707-1783) proved FLT for n=3. But for filling one ‘gap’ that was pointed out in his proof, he introduced algebraic numbers which were not known during Fermat’s time. However, Dr. Gurtu not only filled in this ‘gap’ by introducing hair-splitting type of modifications in Euler’s approach, but extended it to cover the proof of FLT for any other higher odd prime values of n = 5,7,11,…. And all this has been done while remaining in the domain of positive integers. It is really surprising that these simple modifications did not strike any other mathematician including Euler, but then it only goes to prove Dr. Gurtu’s ingenuity which can perhaps be attributed to his Indian origin, something about which French mathematician Pierre Simon de Laplace (1749-1827) has written in following terms – “It is India which gave us the ingenious method of expressing all numbers by means of ten symbols, and the appropriate grandeur of this achievement becomes more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity”.

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