Class Numbers, Quadratics, and Exponential Diophantine Equations

We look at the relationships between class numbers of quadratic structures (orders and fields) and the solutions of exponential Diophantine equations. We conclude with necessary and sufficient conditions for a class group to have an element of a given order. 1. Notation and Preliminaries If D is a squarefree integer, then its discriminant is given by ( ) ( )    ≡ ≡/ = ∆ . 4 mod 1 if , 4 mod 1 if 4 D D D D Then ∆ is called a fundamental discriminant with associated radicand D, and ( ) ( ) ( )     ≡ ∆ ≡ = ∆ + = ω∆ 4 mod 0 if , 4 mod 1 if 2 1 D D D is called the principal surd associated with ∆. This will provide the canonical basis element for our orders. First we need notation for a Z -module R. A. MOLLIN 538 [ ] { }, , : , Z ∈ β + α = β α y x y x where ( ), , ∆ = ∈ β α Q K the real quadratic field of discriminant ∆ and radicand D. For this reason, fundamental discriminants are often called field discriminants. We need to be able to distinguish those Z -modules that are ideals in . ∆ O Theorem 1 (Primitive Ideals and Norms). Let ∆ be a fundamental discriminant, and set ( ) 0 ≠ I be Z -submodule of . ∆ O Then I has a representation of the form [ ], , ∆ ω + = c b a I where N ∈ c a, and Z ∈ b with . 0 a b < ≤ Furthermore, I is an ideal of ∆ O if and only if this representation satisfies , |a c , |b c and ( ). | ∆ ω + c b N ac (For convenience, we call I an ∆ O -ideal.) If , 1 = c then we say that a non-zero ideal I is a primitive ∆ O -ideal. If I is a primitive ∆ O -ideal, then a is the least positive rational integer in I, denoted ( ) a I N = called the norm of I. Proof. See [5, Theorem 1.2.1, p. 9]. We need to know when ideals are equal. Theorem 2 (Criteria for Ideal Equality). If ∆ is a fundamental discriminant and [ ] α = , a I is a primitive ∆ O -ideal, then [ ] α + = za a I , for any integer z. Proof. See [5, Theorem 1.2.2, p. 10]. The class group of ∆ O will be denoted by ∆ C and equivalences of ideal I, J therein by . ~ J I We will also be using the fact that if 1 ~ j I (namely if j I is a principal ideal in ∆ C ), and d is the smallest positive integer such that , 1 ~ d I then . j d | The following will be a useful criterion in our study of complex quadratic ideals in the next section. Theorem 3. If 0 < ∆ is a fundamental discriminant and = I CLASS NUMBERS, QUADRATICS, ... 539 [ ] ∆ ω + b a, is a primitive ∆ O -ideal, satisfying ( ) ( ) , 2 ∆ ∆ ω < ω + N b N then I is principal if and only if either 1 = a or ( ). ∆ ω + = b N a Proof. See [5, Theorem 1.3.2, p. 16], and [5, Exercise 1.5.7, p. 29].