New York Journal of Mathematics

New York J. Math. 8 (2002) 161–168.

New Prime-Producing Quadratic Polynomials Associated with Class Number One or Two

R.A. Mollin

Abstract. This article provides necessary and sufficient conditions for a real quadratic field to have class number one or two in terms of a new set of primeproducing quadratic polynomials

Contents

1. Introduction 161 2. Prime-producers and class number one 163 3. Prime-producers and class number two 165 References 167

1. Introduction

This section is devoted to the elucidation of several facts on ideal theory which we will require in the balance of the paper.

Let D > 1 be a square-free positive integer and set:

σ = \{ \begin{matrix}2 & if D ≡ 1 (mod 4), \\ 1 & otherwise. \end{matrix}

Define ω_{Δ} = (σ − 1 + \sqrt{D})/σ, and Δ = (ω_{Δ} − ω′_{Δ})^{2} = 4D/σ^{2}. The value Δ is called a fundamental discriminant or field discriminant with associated radicand D, and ω_{Δ} is called the principal fundamental surd associated with Δ.

There is a family of discriminants upon which we will concentrate in this paper, defined as follows (see [9] for complete details on their properties and background).

Definition 1.1. If Δ = ℓ^{2} + r is a discriminant with r | 4ℓ, then Δ is said to be of extended Richaud-Degert type (ERD-type).

161^{ISSN 1076-9803/02}

Received April 12, 2002.

Mathematics Subject Classification. 11C08, 11D85, 11R11.

Key words and phrases. polynomials, primes, class numbers, ideals.

The author’s research is supported by NSERC Canada grant # A8484.

Now we develop the notation required for the balance of our discussion. If [α, β] = αℤ + βℤ, then 𝒪_{Δ} = [1, ω_{Δ}] is called the maximal order or ring of integers of K = ℚ(\sqrt{D}). It may be shown that any ℤ-module I ≠ (0) of 𝒪_{Δ} has a representation of the form [a, b + cω_{Δ}], where a, c ∈ ℕ with 0 ≤ b < a. We will only be concerned with primitive ones, namely those for which c = 1. In other words, I is a primitive ℤ-submodule of 𝒪_{Δ} if whenever I = (z)J for some z ∈ ℤ and some ℤ-submodule J of 𝒪_{Δ}, then |z| = 1. Thus, a canonical representation of a primitive ℤ-submodule of 𝒪_{Δ} is obtained by setting σa = Q and b = (P − 1)/2 if σ = 2, while b = P if σ = 1 for P, Q ∈ ℤ, namely

I = [Q/σ, (P + \sqrt{D})/σ]. (1.1)

A nonzero ℤ-module I as given in (1.1) is called a primitive 𝒪_{Δ}-ideal if and only if P^{2} ≡ D (mod Q) (see [11, Theorem 3.5.1, p. 173]). Henceforth, when we refer to an 𝒪_{Δ}-ideal it will be understood that we mean a primitive 𝒪_{Δ}-ideal. Also, the value Q/σ is called the norm of I, denoted by N(I). A reduced ideal I is one which contains an element β = (P + \sqrt{D})/σ such that I = [N(I), β], where β > N(I) and −N(I) < β′ < 0. In fact, the following holds.

Theorem 1.1. If Δ > 0 is a discriminant and I is an 𝒪_{Δ}-ideal with N(I) < \sqrt{Δ}/2, then I is reduced. Conversely, if I is reduced, then N(I) < \sqrt{Δ}.

In particular, for ERD-types, we need to know what the norms of principal reduced ideals happen to be as in the following, each of which can be verified using [9, Theorem 3.2.1, pp. 78–80].

Theorem 1.2. Suppose that Δ = 4(t^{2} ±2) = 4D with t > 3. Then I is a principal, primitive 𝒪_{Δ}-ideal with N(I) < \sqrt{D} if and only if N(I) = 1 or N(I) = 2.

Theorem 1.3. If D = t^{2}+4q ≡ 1 (mod 4) is a fundamental discriminant such that t > 3 and |q| divides t, then I is a primitive principal 𝒪_{D}-ideal with N(I) < \sqrt{D}/2 if and only if N(I) = 1 or N(I) = |q|.

Theorem 1.4. If D = 4t^{2} + q ≡ 1 (mod 4) is a fundamental discriminant with q|t, then I is a primitive principal 𝒪_{Δ}-ideal if and only if N(I) is one of 1, |q|, t + (q − 1)/4, and if q > 0, also t − (q − 1)/4.

We will also need the following theorem on class groups of quadratic fields.

Theorem 1.5. If Δ is the discriminant of a real quadratic field and 𝒞_{Δ} is the class group of ℚ(\sqrt{Δ}), then 𝒞_{Δ} is generated by the non-inert primitive prime ideals with norm less than \sqrt{Δ}/2.

Proof. See [9, Theorem 1.3.1, p. 15]. □

We will also be referring throughout to the exponent of 𝒞_{Δ}, denoted by e_{Δ}, which is the smallest natural number such that I^{e_{Δ}} = 1 for all I ∈ 𝒞_{Δ}. Ideals I, J in the same class of 𝒞_{Δ} are denoted by I ∼ J. The following will also be relevant to our discussion in what follows.

Definition 1.2. If I = [a, (b + \sqrt{Δ})/2] is an 𝒪_{Δ}-ideal, then the ideal I′ = [a, (b − \sqrt{Δ})/2] is called the conjugate of I. When I = I′, I is called an ambiguous ideal, and if I ∼ I′, then I is said to be in an ambiguous class of ideals.

Remark 1.1. It is possible that an ambiguous class of ideals may not contain an ambiguous ideal. This phenomenon and its consequences are explored in detail in [9, Chapter 6, pp. 187–198]. See also the 1989 paper by Louboutin [2].

2. Prime-producers and class number one

This section is devoted to class number one criteria for real quadratic fields of ERD-type in terms of a new family of prime-producing quadratic polynomials. In [12], we provided class number one criteria for arbitrary discriminants in terms of certain prime-producing quadratic polynomials. The results of this section extend those results. However, for ease of exposition, we state our results for only fundamental discriminants. The reader may use the techniques of [12] and the theory developed therein to generalize these results to arbitrary discriminants.

Theorem 2.1. Let Δ = 4D be a fundamental discriminant with radicand D = t^{2} ± 2, for some natural number t > 3. Then h_{Δ} = 1 if and only if

f_{t}(x) = −2x^{2} + 2tx ± 1

is prime for all natural numbers x < t. Also, f_{t}(x) has discriminant Δ.

Proof. To begin, we notice that the prime ideal (t+\sqrt{D}) over the rational prime 2 is principal. If h_{Δ} > 1, then by Theorem 1.5 there exists a non-principle primitive ideal J of odd norm a where 1 < a < \sqrt{D}. Set I = (t + \sqrt{D}) and write I = [2a, b + \sqrt{D}] with −a ≤ b < a. Hence, with −t < b < t and a natural number x with x < t, we may let b = t − 2x, since b ≡ t (mod 2) given that 2a | (b^{2} − D). Since I ≁ 1, then there exists a c > 1 such that N(b + \sqrt{D}) = −2ac, for otherwise we would have I = (b + \sqrt{D}) ∼ 1. Therefore,

2ac = D − b^{2} = −4x^{2} + 4tx ± 2 = 2f_{t}(x),

so f_{t}(x) = ac is composite. We have shown that when f_{t}(x) is prime for all natural numbers x < t, h_{Δ} = 1.

Conversely, if f_{t}(x) = −2x^{2} + 2tx ± 1 is composite for some natural number x < t, then c = c_{1}c_{2} = f_{t}(x) with 1 < c_{1} ≤ c_{2}. Let α = 2x − t + \sqrt{D}, which is primitive with norm N(α) = −2c. Thus, I = [c_{1}, α] is a primitive 𝒪_{Δ}-ideal with N(I) = c_{1}. If c_{1} > \sqrt{D}, then 2c > c > D, so −4x^{2} + 4tx ± 2 > t^{2} ± 2 from which it follows that 0 > (2x − t)^{2}, a contradiction. Hence, by Theorem 1.1, I is reduced. If I ∼ 1, then it follows from Theorem 1.2 that 2c_{1} = 2, contradicting that c_{1} > 1. Hence, I ≁ 1, so h_{Δ} > 1. □

In [13], we provided criteria for arbitrary discriminants to have cyclic subgroups in the class groups of real quadratic orders. The following extends the ideas used therein.

Theorem 2.2. If D = t^{2} ± 2, t > 3, is a fundamental radicand and if there exists an x ∈ ℕ such that −2x^{2} + 2tx ± 1 = c^{n} for some integers c > 1 and n > 1 with gcd(D, c) = 1, then 𝒞_{Δ} has a cyclic subgroup of order n.

Proof. Set α = 2x−t+\sqrt{D}. Then N(α) = −2c^{n} and I = [c, α] is an 𝒪_{Δ}-ideal where Δ = 4D. Since gcd(D, c) = 1, then I^{n} = [c^{n}, α]. Also, since [2, α] = (t + \sqrt{D}) ∼ 1, then I^{n} ∼ 1 ∼ [2c^{n}, α] = [2, α][c^{n}, α]. By the same reasoning as in the above proof, c^{n/2} < \sqrt{D}. If there exists a j | n with j ≠ n such that I^{j} ∼ 1, then since c^{j} ≤ c^{n/2} < \sqrt{D}, I^{j} is reduced. Since t > 3, it follows from Theorem 1.2 that c^{j} = 1, a contradiction. Therefore, the smallest value of j such that I^{j} ∼ 1 is j = n, so 𝒞_{Δ} has a cyclic subgroup of order n. □

Remark 2.1. The above is related to results obtained by this author in 1987 (see [5], as well as [9, Theorem 4.2.4, p. 132], and [15, Theorems 2.2–2.3, p. 475]) as follows. For D = t^{2} ± 2 a fundamental discriminant, let δ be defined by

δ = \{ \begin{matrix}1 & if D ≡ 3 (mod 4), \\ 0 & if D ≡ 2 (mod 4). \end{matrix}

Notice that if we perform the translation x → x + (t − δ)/2 on f_{t}(x) = −2x^{2} + 2tx±1, we get f_{Δ}(x) = −2x^{2}+2δx+(D−δ)/2. The aforementioned result obtained in 1987 is that f_{Δ}(x) is 1 or prime for all x ∈ ℕ with x < (\sqrt{D} + δ)/2 if and only if h_{Δ} = 1. This translates into the result in Theorem 2.1.

In the 1989 publication [16] (see also [9, Conjecture 4.2.1, p. 140]) we posed the following.

Conjecture 2.1 (Mollin-Williams–1989). Let D = pq ≡ 5 (mod 8) where p ≡ q ≡ 3 (mod 4) with q < p are primes. Then the following are equivalent.

(a) |f_{q}(x)| = |qx^{2} + qx + (q − p)/4| is 1 or prime for all nonnegative integers x < \sqrt{D}/4 − 1/2.

(b) h_{D} = 1 and D is of the form D = q^{2}s^{2} ± 4q or D = 4q^{2}s^{2} − q for some s ∈ ℕ.

Remark 2.2. In [3], Louboutin proved that (a) implies h_{D} = 1 and D is of the form D = q^{2}s^{2}±4q if we extend the range of values of x to 0 ≤ x ≤ \sqrt{D}/2−1/2, and states: “This result is our first step towards Mollin-Williams’ conjecture...” Then in [3, Theorem 10] and [4, Theorem 10′] it is proved that if the range of x is 0 ≤ x ≤ \sqrt{D}/3−1 in (a), then (b) holds. However, in [21], Srinivasan (unconditionally) proved that if (a) holds, then h_{D} = 1 and the period length of the simple continued fraction expansion of \sqrt{D} is at most 10. This implies, she notes as a corollary, that Conjecture 2.1 holds with one possible exceptional value of D, whose existence would be a counterexample to the GRH, since for this value of D, (b) would hold but not (a). More recently, in [22], Srinivasan proved (modulo the GRH assumption) that if q is allowed to be any divisor of D and F_{q}(x) = |(p − qx^{2})/4| is 1 or prime for all odd positive integers x < \sqrt{D/5}, then D ≤ 4245. (Note that f_{q}(x) and F_{q}(x) are equivalent since f_{q}(x) = ((2x + 1)^{2}q − p)/4.) This establishes (modulo GRH) a conjecture made by this author in [8, Conjecture 3.1, p. 359] (see also [9, Conjecture 4.2.2, p. 143]). It also shows that if (a) of Conjecture 2.1 holds, then h_{D} = 1 and D is of ERD type (modulo GRH). She also proved, unconditionally, that if q is any divisor of D and F_{q}(x) is 1 or prime for all odd positive integers x < \sqrt{D/5}, then either h_{D} ≤ 2 or D is of ERD type. Under the assumption of the GRH, she proved that if q is any divisor of D and F_{q}(x) is 1 or prime for all odd positive integers x < \sqrt{D/5}, then h_{D} ≤ 2. Another result of interest that she verified (without the GRH assumption) is that if F_{q}(x) is 1 or prime for all odd positive integers x < \sqrt{D}/2 and there exists at least one split prime less than \sqrt{D}/2, then h_{D} ≤ 2 or h_{D} = 4.

A complete list (with one possible GRH-ruled-out exception) of ERD-types having class number one is given in [9, Theorem 5.4.3, p. 176], which first appeared in the 1990 publication [17], although the announcement of it was made in a note added in proof at the end of the 1987 paper [6].

We address the class number two problem in the next section.

3. Prime-producers and class number two

We begin with a result that provides necessary and sufficient conditions for h_{Δ} ≤ 2 and extends the result in [7, Proposition 3.1, p. 89] where continued fraction techniques were used.

Theorem 3.1. Let Δ = 4D = 4(t^{2} ± 2) be a fundamental discriminant, set

𝒮 = \{odd primes p < \sqrt{D} : (D/p) ≠ −1\}

where the symbol on the right is the Legendre symbol, and let

f_{t}(x) = −2x^{2} + 2tx ± 1.

Then the following are equivalent.

(a) Either h_{Δ} = 1 and 𝒮 = ∅, or h_{Δ} = 2.

(b) For each p ∈ 𝒮 there exists an x ∈ ℕ with x ≤ t/2 such that f_{t}(x) = pr_{p} where r_{p} is a prime which is the norm of 𝒪_{Δ}-ideal ℛ_{p} in an ambiguous class and ℛ_{p} ∼ ℛ_{p′} for all p, p′ ∈ 𝒮.

Also, f_{t}(x) has discriminant Δ.

Proof. Given that the result is vacuously true when t ≤ 3 (since in those cases h_{Δ} = 1 and 𝒮 = ∅), we will assume throughout that t > 3.

First suppose that (b) holds. If h_{Δ} = 1, then by Theorem 2.1, 𝒮 = ∅. Thus, we may assume that h_{Δ} > 1. By Theorem 1.5, 𝒞_{Δ} is generated by the non-inert primitive prime ideals 𝒫_{p} with N(𝒫_{p}) = p < \sqrt{D}. Since [2, t+\sqrt{D}] = (t+\sqrt{D}) ∼ 1, then we may assume that p ∈ 𝒮. Given such a p, the hypothesis tells us that there exists a natural number x ≤ t/2 such that f_{t}(x) = pr_{p} for some prime r_{p} which is the norm of an ambiguous 𝒪_{Δ}-ideal. If we set b = t − 2x and consider [2r_{p}p, b + \sqrt{D}] = (b + \sqrt{D}) ∼ 1, then 𝒫_{p}ℛ_{p} ∼ 1 where ℛ_{p} = [2r_{p}, b + \sqrt{D}] and 𝒫_{p} = [p, b + \sqrt{D}]. Therefore, by (b), ℛ_{p} ∼ ℛ′_{p} ∼ 𝒫_{p}ℛ_{p}ℛ′_{p} ∼ 𝒫_{p}. Since (b) also tells us that ℛ_{p} ∼ ℛ_{p′} for all p, p′ ∈ 𝒮, then 𝒫_{p} ∼ 𝒫_{p′} for all p, p′ ∈ 𝒮. Moreover, since ℛ_{p}^{2} ∼ 1 for each p ∈ 𝒮, then 𝒫_{p}^{2} ∼ 1 for all p ∈ 𝒮. Hence, h_{Δ} = 2 (since the only ideals in the principal class of 𝒞_{Δ} with norms less than \sqrt{D} are the trivial ideal and the ideal over 2 by Theorem 1.2, which we may invoke since t > 3). We have shown that (b) implies (a).

Conversely, we may assume that (a) holds and that 𝒮 ≠ ∅ since the result is vacuously true otherwise. Thus, h_{Δ} = 2. Let p ∈ 𝒮 and set [2p, b + \sqrt{D}] ∼ [p, b + \sqrt{D}] = 𝒫 with 0 ≤ b < p, which is a generator of 𝒞_{Δ} by Theorem 1.5. Since p < \sqrt{D}, then b < 2t, so we may set b = t−2x where 1 ≤ x ≤ t. Since 𝒫 ≁ 1, then as in previous arguments, there exists a c > 1 such that 2pc = D−b^{2} = −4x^{2}+4xt±2, so pc = −2x^{2} + 2xt ± 1 = f_{t}(x).

We now show that c is prime. If c = c_{1}c_{2} with 1 < c_{1} ≤ c_{2}, then we may set α = 2x − t + \sqrt{D} and it follows that I = [c_{1}, α] is a primitive 𝒪_{Δ}-ideal with N(I) = c_{1}. If c_{1} > \sqrt{D}, then c_{1}c_{2} = c ≥ c_{1}^{2} > D. However, D ≥ 2f_{t}(x), so c > 2pc, a contradiction. Thus, c_{1} < \sqrt{D} which means that I is reduced by Theorem 1.1.

If I ∼ 1, then by Theorem 1.2, c_{1} = 1 or c_{1} = 2, both of which are contradictions since c_{1} > 1 by choice and c is odd. Hence, I ≁ 1. Since [pc, α] ∼ 1, then

[p, α] ∼ [c, α], (3.2)

given that h_{Δ} = 2. Since [c, α] = [c_{1}, α][c_{2}, α], then if [c_{2}, α] ≁ 1, we must have that [c, α] ∼ 1 since h_{Δ} = 2. However, this contradicts that 𝒫 ≁ 1 in view of (3.2). Hence, [c_{2}, α] ∼ 1. Assume that c_{2} > \sqrt{D}. If pc_{1} > \sqrt{D}, then D < pc_{1}c_{2} = pc = f_{t}(x) ≤ D/2, a contradiction, so pc_{1} < \sqrt{D}. Given that 𝒫 ≁ 1, [c_{1}, α] ≁ 1, and h_{Δ} = 2, then [pc_{1}, α] ∼ 1. Thus, by Theorems 1.1–1.2, pc_{1} ∈ {1, 2}, a contradiction. Hence, c_{2} < \sqrt{D} and [c_{2}, α] ∼ 1, so as above, c_{2} = 1 or c_{2} = 2, both of which are contradictions. We have therefore shown that c = r_{p} is prime. By (3.2), ℛ_{p} = [r_{p}, α] ∼ 𝒫. Since t > 3, we may invoke Theorem 1.2 which tells us that no element of 𝒮 can be the norm of a principal 𝒪_{Δ}-ideal, this then is sufficient to show that (a) implies (b). □

Remark 3.1. In 1991, we established class number 2 criteria for ERD-types in [7], using the polynomial f_{Δ}(x) defined in Remark 2.1, which involved simple continued fraction expansions of quadratic irrationals. However, the criterion did not include the types D = t^{2} ± 2, which were problematic for the continued fraction approach. Recently, in [14], we were able to provide such continued fraction criteria for the latter types as well as some new such criteria for other ERD-types. However, some of the ERD-types were also excluded in [14], namely those of the form D = t^{2} ±2 = 2p where p is prime. The reason is that these types are those for which there exist ambiguous classes of ideals in 𝒪_{Δ} without ambiguous ideals in them. The approach given in Theorem 3.1 does not suffer from this defect and so is more general. Thus, as with the class number one criteria given in the previous section, we have new class number two criteria in terms in the polynomials f_{t}(x).

In [14], we posed the conjecture that if Δ = 4(t^{2} ± 2) = 4D is a fundamental discriminant, then h_{Δ} = 2 if and only if D is one of 34, 66, 102, 119, 123, 146, 194, 258, 287, 402, 482, 527, 623, 678, 782, 843, 902, 1022, 1298. Moreover, given the aforementioned techniques, we know that the list is complete with one GRH-ruled-out exception. Given the comments in Remark 1.1, it is worthy of note that the only values of D in the aforementioned list where the class group is generated by an ambiguous class without ambiguous ideals are D ∈ {34, 146, 194, 482}.

We now look at criteria for class number two for the remaining ERD-types in terms of quadratic polynomials which behave in a fashion similar to Theorem 3.1. We do not provide proofs for the following since the arguments are analogous to those presented above by using Theorems 1.3–1.4.

Theorem 3.2. If Δ = t^{2} + 4q ≡ 5 (mod 8) is a fundamental discriminant with q | t, and 𝒮 = \{primes p ≠ |q| < \sqrt{Δ}/2 : (Δ/q) ≠ −1\}, then the following are equivalent.

(a) Either h_{Δ} = 1 and 𝒮 = ∅ or h_{Δ} = 2.

(b) For all p ∈ 𝒮 there exists a natural number x < t/2 such that

|f_{t}(x)| = | − x^{2} + tx + q| = pr_{p},

where r_{p} is a prime that is the norm of an ideal ℛ_{p} in an ambiguous class of 𝒞_{Δ} and ℛ_{p} ∼ ℛ_{p′} for all p, p′ ∈ 𝒮.

New Prime-Producing Quadratic Polynomials 167

Theorem 3.3. If Δ = 4t^{2} + q ≡ 1 (mod 4) is a fundamental discriminant with |q| | t and

𝒮 = \{primes p < \sqrt{Δ}/2 : p ≠ |q|, t ± (q − 1)/4, and (Δ/p) ≠ −1\},

then the following are equivalent.

(a) Either h_{Δ} = 1 and 𝒮 = ∅ or h_{Δ} = 2.

(b) For all p ∈ 𝒮 there exists a natural number x ≤ t such that

f_{t,q}(x) = −x^{2} + +(2t + 1)x + (q − 4t − 1)/4 = pr_{p},

where r_{p} is a prime that is the norm of an ideal ℛ_{p} in an ambiguous class of 𝒞_{Δ} and ℛ_{p} ∼ ℛ_{p′} for all p, p′ ∈ 𝒮.

Also, f_{t,q}(x) has discriminant Δ.

In the 1991 publication [18] (see also [9, Table A9, p. 286]) we established the complete list of ERD type discriminants Δ with h_{Δ} = 2 (with one GRH-ruled out exception). The largest of these is Δ = 14405 and the second largest is Δ = 9005. Also, in [19]–[20], we established under the assumption of the GRH that if Δ is any fundamental discriminant, h_{Δ} ≤ 2, and the period length of the simple continued fraction expansion of ω_{Δ} is at most 25, then Δ ≤ 248093 (see also [9, Table A3, pp. 274–277]).

Acknowledgements. We thank the referee who made comments that resulted in a more concise and polished version of the paper.

References

[1] A. Granville, R.A. Mollin, and H.C. Williams, An upper bound on the least inert prime in a real quadratic field, Can. J. Math. 52 (2000), 369–380, MR 2001d:11123.

[2] S. Louboutin, Groupes des classes d’Iéaux triviaux, Acta Arith. 54 (1989), 61–74, MR 91a:11051, MR 0634.12008.

[3] S. Louboutin, Prime producing quadratic polynomials and class-numbers of real quadratic fields, Can. J. Math. 42 (1990), 315–341, MR 91f:11073, Zbl 0711.11041.

[4] S. Louboutin, Addendum to: Prime producing quadratic polynomials and class-numbers of real quadratic fields, Can. J. Math. 42 (1990), 1131, MR 91m:11093, Zbl 0726.11065.

[5] R.A. Mollin, Class number one criteria for real quadratic fields II, Proceed. Japan Acad. Sci., Ser. A, 63 (1987), 162–164, MR 88k:11073b, Zbl 0625.12003.

[6] R.A. Mollin, An overview of the solution to the class number one problem for real quadratic fields of Richaud-Degert type, Number theory, Vol. II (Budapest, 1987) Colloquia Math. Soc. János Bolyai, 51, North Holland, Amsterdam, 1990, 871–888, MR 91e:11120, Zbl 0702.11071.

[7] R.A. Mollin, Applications of a new class number two criterion for real quadratic fields, Computational Number Theory (Debrecen, 1989), Walter de Gruyter, Berlin, New York (1991), 83–94, MR 93e:11133, Zbl 0734.11057.

[8] R.A. Mollin, Ambiguous classes in quadratic fields, Math. Comp. 61 (1993), 355–360, MR 93k:11103, Zbl 0790.11076.

[9] R.A. Mollin, Quadratics, CRC Press, Boca Raton, New York, London, Tokyo, 1996, MR 97e:11135, Zbl 0858.11001.

[10] R.A. Mollin, Prime-producing quadratics, Amer. Math. Monthly, 104 (1997), 529–544, MR 98h:11113, Zbl 0886.11053.

[11] R. A. Mollin, Fundamental Number Theory with Applications, CRC Press, Boca Raton, New York, London, Tokyo, 1998, Zbl 0943.11001.

[12] R.A. Mollin, Class number one and prime-producing quadratic polynomials revisited, Can. Math. Bull. 41 (1998), 328–334, MR 99g:11127, Zbl 0920.11078.

[13] R.A. Mollin, Cyclic subgroups of ideal class groups in real quadratic orders, Glasgow Math. J., 41 (1999), 197–206, MR 2000i:11164, Zbl 0990.11066.

168 R.A. Mollin

[14] R.A. Mollin, Continued fractions and class number two, Int. J. Math. and Math. Sci. 27 (2001), 565–571, MR 2002k:11190.

[15] R.A. Mollin and H.C. Williams, Quadratic non-residues and prime-producing polynomials, Can. Math. Bull. 32 (1989), 474–478, MR 91a:11009, Zbl 0714.11066.

[16] R.A. Mollin and H.C. Williams, Prime-producing quadratic polynomials and quadratic fields of class number one, Théorie des Nombres (Quebec, PQ, 1987) (J.-M. DeKoninck and C. Levesque, eds.), Walter de Gruyter, Berlin, 1989, 654–663, MR 90m:11153, Zbl 0695.12002.

[17] R.A. Mollin and H.C. Williams, Solution of the class number one problem for real quadratic fields of extended Richaud-Degert type (with one possible exception), Number Theory (Banff, AB, 1988) (R.A. Mollin, ed.), Walter de Gruyter, Berlin, 1990, 417–425, MR 92f:11144, Zbl 0696.12004.

[18] R.A. Mollin and H.C. Williams, On a solution of a class number two problem for a family of quadratic fields, Computational Number Theory (Debrecen, 1989), Walter de Gruyter, Berlin, New York, 1991, 95–101, MR 93d:11118, Zbl 0734.11058.

[19] R.A. Mollin and H.C. Williams, On a determination of real quadratic fields of class number two and related continued fraction period length less than 25, Proceed. Japan Acad. 67, Ser. A (1991), 505–516.

[20] R.A. Mollin and H. C. Williams, On real quadratic fields of class number two, Math. Comp. 59 (1992), 625–632, MR 93a:11089, Zbl 0774.11061.

[21] A. Srinivasan, Prime producing polynomials: Proof of a conjecture by Mollin and Williams, Acta Arith. 89 (1999), 1–7, MR 2000h:11112, Zbl 0927.11051.

[22] A. Srinivasan, Prime producing quadratic polynomials and class number one or two, (to appear).

Mathematics Department, University of Calgary, Calgary, Alberta, T2N 1N4, Canada ramollin@math.ucalgary.ca http://www.math.ucalgary.ca/˜ ramollin/

This paper is available via http://nyjm.albany.edu:8000/j/2002/8-10.html.