Of Q-quasi-prime Cyclotomic Units And Diophantine Equations

Preda Mihailescu introduced the concept of q-primary cyclotomic units C_q of Q(ζ_p)⁺ in proving Catalan conjecture, and studied the structure of C_q which played an important role in his proof. In this paper we obtain some results about the corresponding q-primary cyclotomic units group on Q(ζ_pⁿ)⁺ as follows:(1) Let C denote the cyclotomic units group of Q(ζ_pⁿ)⁺, then C/C_q is a cyclic F_q[G⁺]/(N)-module. Furthermore, it has full support.(2) F_q[G⁺]-module C/C_q has a direct sums decomposition: C/C^q = C/C_q(?) C_q/C^q.(3) If C = C_q, then q > (?)(pⁿ).In the second part of this paper we give a necessary condition to determine whether the equation x^p - y² = l has integral solutions with p > [l/2]. In the last part, we give solutions for some Diophantine equations relating to the rational, torsion subgroups of elliptic curves over Q.