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=cⁿn^kn-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