Retracted: A Proof of Fermat's Last Theorem using an Euler's Equation

Abstract

Fermat's Last Theorem states that there are no solutions to xn + yn = zn for n ≥ 3 and x; y; z non-zero integers. Fermat wrote down a proof for n = 4 [1]. In 1753, Lenohard Euler (1707{1783) wrote down a proof of FLT for the exponent n = 3 [1]. Since any integer n ≥ 3 is divisible by an odd prime number or by 4, it is sucient to prove FLT for n = p, an odd prime > 3. We prove the theorem for p ≥ 5. We consider x3 + y3 = z3 and sp + tp = up, where p is any prime > 3. Without loss of generality it is enough to assume that both x and y as non-zero positive integers; therefore z3 will be a non-zero positive integer, but both z and z2 will be irrational in the rst equation. We hypothesize that there exist positive integers, s; t and u in the Fermat's equation sp + tp = up and bring a contradiction. We have created by trial and error method two equivalent equations to Fermat's equations. x3 + y3 = z3 and sp + tp = up through parameters a; b; c; d; e and f given by respectively. The values of a; b; c; d; e and f in terms of known values are worked out in the detailed proof. Also we can obtain the equivalent values of 2n; 7k2 and l2 from these equations where 2n = 7k2 + l2. (The validity of the equivalent equations has been discussed in detail in Annexure{A). Solving the above two equivalent equations using the above Euler's equation, we arrive at the result stu = 0, thus proving the theorem. Use of the Euler's equation leads us to this elementary proof. Retraction Notice: This paper has been retracted from the journal after receipt of written complains. This journal is determined to promote integrity in research publication. This retraction is in spirit of the same. After formal procedures editor(s) and publisher have retracted this paper on 18th September-2018. Related policy is available here: http://goo.gl/lI77Nn