of whether it is possible for a number to satisfy the congruences of Wieferich and Mirimanoff simultaneously. To date, the only known Mirimanoff primes are 11 and 1006003 . The discovery of the second of these appears to be due to K.E. Kloss (1965). References K.E. Kloss, "Some Number-Theoretic Calculations," Journal of Research of the National Bureau of Standards—B. Mathematics and Mathematical Physics 69 (1965), pp. 335–336. Emma Lehmer, "On Congruences involving Bernoulli Numbers and the Quotients of Fermat and Wilson," Annals of Mathematics 39 (1938), pp. 350–360. M. Lerch, "Zur Theorie des Fermatschen Quotienten…," Mathematische Annalen 60 (1905), pp. 471–490 . D. Mirimanoff, "Sur la Congruence (rp−1 − 1):p ≡ qr," Journal für die reine und angewandte Mathematik 115 (1895), pp. 295–300 . Some corrections are given in the 1937 paper below. D. Mirimanoff, "Sur le dernier théorème de Fermat et le Critérium de M. A. Wieferich," L'Enseignement Mathématique 11 (1909), pp. 455–459 . D. Mirimanoff, "Sur le dernier théorème de Fermat," Comptes rendus hebdomadaires des séances de l'Académie des Sciences 150 (1910), pp. 204–206; a revised and expanded version of this paper appeared under the same title in Journal für die reine und angewandte Mathematik 139 (1911), pp. 309–324 . D. Mirimanoff, "Sur les nombres de Bernoulli," L'Enseignement Mathématique 36 (1937), pp. 228–235 . Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979 Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer, 2006 Number theory