ELEMENTARY PROOFS FOR THE FERMAT'S LAST THEOREM IN Z USING ONE TRICK FOR A RESTRICTION IN ZP

Abstract

An elementary and short proof of Fermat's Last Theorem (FLT) is presented, which is understandable even to a student. Perhaps this proof is precisely the lost proof, which could similar to own Fermat's proof. Restricting some coefficients of polynomials by value 0, except for the first term, allows to prove the Fermat's Last Theorem for domain Z, since in this case the canonical representation of p-adic numbers is limited to only one digit in the corresponding p-ary system. It was shown within the framework of elementary algebra, which corresponds to the Pythagorean theorem (PT) that the assumption of the existence of certain “Fermat’s triples” (FT), as integer solutions of Fermat's Last Theorem, can not be possible in Z due to some fatal inconsistencies for the PT and found by means the PT. Some equations in Zp were shown for n=3, 4 and 5