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An Introduction to Fermats Last Theorem Fermat claimed to have found a proof of the theorem at an early stage in his career Much later he spent time and effort proving the cases n4 and n5 Had he had a proof to his theorem earlier there would have been no need for him to study specific cases It is likely that he found a mistake in his own proof before he had a chance to announce the result and never bothered to erase the marginal comment because it never occurred to him that anyone would see it there1 Fermats Last Theorem was not actually an important landmark in the development of mathematics but has precipitated considerable research of significant impact on number theory and algebraic geometry 2 It was only finally solved by a mildly obsessed professor who had a personal struggle with the problem There was virtually no money to be gained by its proof and generated little interest outside the mathematical fraternity Analysis Fermat may have had the following proof in mind when he wrote his famous comment 1 Fermat discovered and applied the method of infinite descent which in particular can be used to prove Fermats Last Theorem for n4 This method can actually be used to prove a stronger statement than Fermats Last Theorem for n4 viz x4 y4 z2 has no non-trivial integer solutions It is possible and even likely that he had an incorrect proof of Fermats Last Theorem using this method when he wrote the famous theorem The incorrect proof probably goes something like this It is true that only the prime exponents need to be considered So consider xn yp zp Let r be a primitive pth root of unity complex number Then the equation is the same as x yx ryx r2yx rp - 1y zp Now consider the ring of the form
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