Logical proofs for Fermat’s Last Theorem

With the help of elliptic curve, Gerardin [4] derived a general expression for the equal sum of two, two fourth powers. It gives a way to prove FLT logically.

(a + 3a² – 2a³  + a⁵  + a⁷)⁴  + ( 1 + a² -2a⁴ -3a⁵  + a⁶ )⁴  =  (a - 3a² – 2a³  + a⁵  + a⁷)⁴  + ( 1 + a² -2a⁴ +3a⁵  + a⁶ )⁴

If one of the members is made to be equal to zero, the remaining relation will not be true for any integral values. The conditions  1 ± 3a + a⁴  + a⁶  = 2a²  due to disparity of odd-evenness and 1 = a² ( 2a² ± 3a³  - 1 – a⁴)  give only impractical solutions which can be considered as a confirmation of FLT.