A new condition equivalent to the Ankeny-Artin-Chowla conjecture

Let p≡1(mod4) be prime, and let ϵ=(t+up)/2 be the fundamental unit of Q(p). In 1952, Ankeny, Artin and Chowla asked if ϵ always has the property that u≢0(modp). The conjecture that the answer to this question is affirmative is known as the Ankeny-Artin-Chowla (AAC) conjecture, and is still unresolved. In this article, we present a new condition that is equivalent to the AAC-conjecture. Additionally, we provide a similar condition that is equivalent to the analogous conjecture of Mordell for the case when p≡3(mod4). Both of these conditions involve certain Lucas polynomials. Moreover, using theorems of Capelli, we provide a different approach to establish the sufficiency of these polynomial conditions.