Let p be an odd prime and suppose that for some a, b, c ϵ Z \\ p Z we have that a p + b p + c p = 0. In Part I a simple new expression and a simple proof of the congruences of Mirimanoff which appeared in his papers of 1910 and 1911 are given. As is known, these congruences have Wieferich and Mirimanoff criteria (2 p − 1 ≡ 1 mod p 2 and 3 p − 1 ≡ 1 mod p 2) as immediate consequences. Mirimanoff's congruences are expressed in the form of polynomial congruences P m( −a b ) ≡ 0 mod p , 1 ≤ m ≤ p − 1, and these polynomials P m ( X) are characterized by means of simple relations. In Part II a complement to Kummer-Mirimanoff congruences is given under the hypothesis that p does not divide the second factor of the class number of the p-cyclotomic field.