Let d be a positive square-free integer and 𝜖d = (Td + Udd)/2 > 1 be the fundamental unit of the real quadratic field ℚ(d). The Ankeny–Artin–Chowla (AAC) conjecture asserts that Up≢0 (mod p) for primes p ≡ 1 (mod 4), which still remains unsolved. In this paper, sufficient conditions for Ud < d have been given in terms of Yokoi’s invariants nd and md, and it has been shown that the AAC conjecture is true in some special cases.

中文翻译：

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横井不变量和 Ankeny-Artin-Chowla 猜想

让d是一个正的无平方整数，并且𝜖d = (吨d + üdd)/2 > 1是实二次域的基本单位ℚ(d). Ankeny-Artin-Chowla (AAC) 猜想断言üp≢0（模组p) 对于素数p ≡ 1（mod 4），仍然没有解决。在本文中，充分条件üd < d已经根据 Yokoi 的不变量给出nd和米d, 并且已经证明 AAC 猜想在某些特殊情况下是正确的。