Factorisation dans un Ordre Non Maximal d'un Corps Quadratique

Let O_f be an order of index f in a quadratic field. We denote A_f the set of elements of O_f whose norm is relatively prime to f. An element v ∈ A_f is called k-prime if for x, y ∈ A_f, v|xy implies v|x^k or v|y^k where k is the exponent of the group O_f. We prove that the k-th powers of the elements of A_f have a unique representation as a product of elements which are irreductible and k-prime. One criteria for resolution of some diophantine equations is an illustration from it.