First case of Fermat's Last Theorem

The first case of Fermat's last theorem says that for three integers x, y and z and a prime number p, where p does not divide the product xyz, there are no solutions to the equation x^p + y^p + z^p = 0.

Using the Theorem of unique factorization of ideals in Q(ξ) it was shown that if the first case has solutions x, y, z, then x+y+z is divisible by p and (x, y), (y, z) and (z, x) are elements of H_p, where H_p denotes a set of pairs of integers with special properties.