14 d (including d = 2), and that every solution has the form ± ( x$_{n}$, y$_{n}$ ) where x$_{n}$ + √$_{dy}$$_{n}$ = ( x$_{1}$ $_{+}$√$_{dy}$$_{1}$$_{)}$ n for some solution ( x$_{1}$, y$_{1}$ ) and n ∈ Z . Acknowledgment Many thanks to Professor Dan Segal, All-Souls College, Oxford, for his advice. References [1] Borevich, Z. I., and Shafarevich I. R., Number Theory, Academic Press, New York, 1973. [2] C. S. Dalawat, Primary units in cyclotomic fields, Annales des sciences math´ ematiques du Qu´ ebec to appear, 2011. [3] G. L. Dirichlet, Sur la mani`ere de r´esoudre l’´equation t 2 − pu 2 = 1 au moyen des fonctions circulaires, Journal f¨ ur die reine und angewandte Mathematik 17, pp. 286-290, 1837. [4] V. Flynn, Algebraic Number Theory Lecture Notes . University of Oxford. Oxford Mathematical Institute, Oxford, UK. 2011. Lecture Notes. [5] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1982. [6] S. Lang, Algebraic Number Theory, Springer-Verlag, New York, 1986. [7] L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982. KABALAN GASPARD