Modular Arithmetic on Elements of Small Norm in Quadratic Fields

Abstract

We describe an algorithm which rapidly computes the coefficients of elements of small norm in quadratic fields modulo a positive integer. Our method requires that an approximation of the natural logarithm of that quadratic field element is known to sufficient accuracy. To demonstrate the efficiency and utility of our method, we apply it to eliminate a number of exceptional cases of a theorem of Dujella and Petho l9r involving Diophantine triples. In particular, we are able to show that Theorem 1.2 of l9r is unconditionally true for all k ≤ 100 with the possible exception of k e 37, for which the theorem holds under the assumption of the Extended Riemann Hypothesis.