Fermat-Like Binomial Equations

Abstract

For more than three centuries it has been a conjecture that the Diophantine equation

$$ {x^n} + {y^{_n}} = {z^n} $$

((1))

has no solutions in natural numbers x, y, z for all n > 2. At present this conjecture, which is also called ”Fermat’s Last Theorem”, is known to be true for all n ≤ 125 000 [1]. Moreover, the recent work of G. Faltings (see [1]) implies that, for each n ≥ 3, (1) has at most a finite number of solutions (x, y, z), with (x, y, z) = 1 and xyz ≠ 0.