An Elementary Proof for Fermat’s Last Theorem using Ramanujan–Nagell Equation

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P. N. Seetharaman

Abstract

 Fermat’s  Last  Theorem  states  that  it  is  impossible  to  find  natural  numbers  A, B  and  C


satisfying the equation


An  + Bn  = Cn


where n is any integer > 2.


Taking  the  proofs  of  Fermat  for  the  index  n =  4,  and  Euler  for  n =  3,  it  is  sufficient  to


prove the theorem for n = p, any prime > 3 [1].


We hypothesis that all rs and t are non-zero integers in the equation


rp  + sp  = tp


and establish contradiction, and assert that r is a non-zero integer.


Just for supporting the proof in the above equation, we have another equation


x3  + y3  = z3


Without loss of generality, we assert that both x and y as non-zero integers; z3  a non-zero integer; z  and z2  irrational.


By trial and error method, we create the transformed equations to the above two equations, solving by which we prove the the theorem.

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