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Abstract

Fermat's last theorem was proposed more than 350 years ago, it attracted the interests of a lot of researchers [1][2][3][4][5][6][7][8][9][10][11] .The simplest case of Fermat's last theorem is n=3, but the previous proofs on it are generally complex or not easy to understand.The present work through the transformation x=t+1, firstly proves that when the values of x and t{ tmin, tmax}  { xmin, xmax }, the Fermat's last theorem in the case of n=3 is true.Furthermore, the paper also proves that when x take other positive integers besides x{ tmin, tmax}  {xmin, xmax}, the Fermat's last theorem in the case of n=3 is true as well.Therefore, there are no a group of positive integers of x, y and z to satisfy the Diophantine equation in the case of n=3; Fermat's last theorem in the case of n=3 is true.