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

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

Journal Title: Asian Research Journal of Mathematics - Year 2017, Vol 6, Issue 3

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( a √ √u + b 2n=2 )2 + ( a − b √ l5=3 √ 23n=2 )2 = ( e √ 71=3k5=3 + f )2 ( c √ √ xt + d 75=3k7=3 )2 + ( c √ y − d √ l7=3 )2 = ( e − f √ z )2 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.

Authors and Affiliations

P. N. Seetharaman

Keywords

a; b; c; d; e; f parameters in the equivalent equations to Fermat's equations x3 + y3 = z3 and sp + tp = up; 2n = 7k2 + l2.

EP ID EP338460
DOI 10.9734/ARJOM/2017/36405
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