Lemma. In a number system with a prime base q [below q>Cn] for single-digit positive number r (q>r>0) there exists such a number g that rg≡1 (mod q); [=> C=de].
Let for the lowest C, relatively prime naturals numbers A, B, C and prime n>2
1°) An+Bn=Cn [=(A+B)R], where, as is known,
1a°) either A+B=cn and R=rn, or A+B=cnnkn-1 and R=nrn, where c>0 and r>0.
An elementary proof of the Fermat’s Last Theorem
2°) After multiplying the equality 1° by gn, where rg≡1 (mod q), on the last digits of
degrees (Ag)n, (Bg)n, (Cg)n we obtain a double of the Fermat’s equality with cn<Cn:
3°) an+bn=cn (anything else is excluded!), which proves the Last Theorem.
Mezos. March 4, 2014