Rings of integral quaternions

A quaternion order derived from an integral ternary quadratic form f = Σa_ijx_ix_j of discriminant d = 4 det (a_ij) is m-maximal if m is not divisible by any prime p such that p² | d, or p 6; d and c_p = 1. If R is m-maximal and m is a product p₁ … p_r of primes, then any primitive element α of R has unique right-divisor ideals of each norm p₁ … p_k (k = 1, …, r). This generalizes Lipschitz's ninety-year-old theorem. We characterize m-maximal orders, study their ideals, and show how the preceding result yields formulas for the number of representations of integers by certain quaternary quadratic forms.