Lehmer's equations in new domains

Abstract

In this paper, we examine the equations kφG(β) = q(β)±1 and kφG(β) = q(β) in the domains of Gaussian integersZ[i] and the polynomial rings over finite fieldsZp[x], where p is prime inZ. Properties concerning the existence of solutions are studied. Complete characterization for the solutions of the above equations is given in the two domains.