The Rabinowitsch-Mollin-Williams Theorem Revisited

Abstract

We completely classify all polynomials of type (𝑥²+𝑥−(Δ−1))/4 which are prime or 1 for a range of consecutive integers 𝑥≥0, called Rabinowitsch polynomials, where Δ≡1(mod4) with Δ>1 square-free. This corrects, extends, and completes the results by Byeon and Stark (2002, 2003) via the use of an updated version of what Andrew Granville has dubbed the Rabinowitsch-Mollin-Williams Theorem—by Granville and Mollin (2000) and Mollin (1996). Furthermore, we verify conjectures of this author and pose more based on the new data.

1. Introduction

The renowned Rabinowitsch result for complex quadratic fields proved in 1913, published in [1], says that if Δ=1−4𝑚 is square-free, then the class number, ℎ_Δ, of the complex quadratic field ℚ(√Δ) is 1 exactly when 𝑥²+𝑥+𝑚 is prime for all integers 𝑥∈[0,𝑚−3]. The Rabinowitsch-Mollin-Williams Theorem is the real quadratic field analogue of the Rabinowitsch result, introduced in 1988, published in [2] by this author and Williams. In [2] and in subsequent renderings of the result, we considered all values of Δ. However, the case where Δ≢1(mod4) is essentially trivial, and the values (unconditionally) known for these Rabinowitsch polynomials are Δ∈{2,3,6,7,11}—see [3]. Therefore, we consider only the interesting case, namely, Δ≡1(mod4).

Theorem 1.1 (Rabinowitsch-Mollin-Williams). If Δ=1+4𝑚, where 𝑚∈ℕ, then the following are equivalent. (a)𝑓_𝑚(𝑥)=𝑥²+𝑥−𝑚 is 1 or prime for all integers 𝑥∈[1,√𝑚]. (b)ℎ_Δ=1 and Δ=𝑠²+𝑟, where 𝑟∈{1,±4}.

Proof. See [2], as well as [4, 5] and Theorem 3.14 below for an update.

A version of Theorem 1.1 was rediscovered by Byeon and Stark [6] in 2002. Then in 2003 [7], they claimed to have classified all of the Rabinowitsch polynomials. However, their list is incomplete. In this paper, we provide the complete and unconditional solution of finding all Rabinowitsch polynomials of narrow Richaud-Degert type, namely, those for which Δ=𝑠²+𝑟 where 𝑟∈{±1,±4}, adding three values missed in [7]. The balance of the Rabinowitsch polynomials turn out to be of wide Richaud-Degert type, namely for those of the form Δ=𝑠²+𝑟, where 𝑟∣4𝑠. In this case, we cite the well-known methodology for showing that the balance of the list is complete “with one possible GRH-ruled-out exception" and add two values missed in [7]. (Here GRH means the generalized Riemann hypothesis.) Lastly, we show how four conjectures posed by this author in 1988 in [8] are affirmatively settled via the above and complement another conjecture by this author affirmatively verified by Byeon et al. in [9].

2. Preliminaries

We will be discussing continued fraction expansions herein for which we remind the reader of the following, the details and background of which may be found in [10] or for a more advanced approach in [4].

We denote the infinite simple continued fraction expansion of a given 𝛼∈ℝ by 𝛼=⟨𝑞₀;𝑞₁,𝑞₂,…⟩where𝑞_𝑗∈ℕfor𝑗∈ℕ,𝑞₀=⌊𝛼⌋, (2.1) where ⌊𝛼⌋ is the floor of 𝛼, namely, the greatest integer less than or equal to 𝛼. It turns out that infinite simple continued fraction expansions are irrational. There is a specific type of irrational that we need as follows.

Definition 2.1 (quadratic irrationals). A real number 𝛼 is called a quadratic irrational if it is an irrational number which is the root of 𝑓(𝑥)=𝑎𝑥²+𝑏𝑥+𝑐, where 𝑎,𝑏,𝑐∈ℤ and 𝑎≠0.

Remark 2.2. A real number 𝛼 is a quadratic irrational if and only if there exist 𝑃,𝑄,Δ∈ℤ such that 𝑄≠0, Δ∈ℕ is not a perfect square, and √𝛼=𝑃+Δ𝑄,(𝑃,𝑄∈ℤ). (2.2) Moreover, if 𝛼 is a quadratic irrational, then 𝑄∣(𝑃²−Δ). Also, 𝛼^′=(𝑃−√Δ)𝑄 (2.3) is called the algebraic conjugate of 𝛼. Here both 𝛼 and 𝛼^′ are the roots of 𝑓(𝑥)=𝑥²−Tr(𝛼)𝑥+𝑁(𝛼), (2.4) where Tr(𝛼)=𝛼+𝛼^′ is the trace of 𝛼, and 𝑁(𝛼)=𝛼⋅𝛼^′ is the norm of 𝛼—see [10, Theorem  5.9, page 222].

Now, given a quadratic irrational 𝛼=(𝑃+√Δ)/𝑄, set 𝑃=𝑃₀, 𝑄=𝑄₀, and for 𝑗≥0 define 𝑃_𝑗+1=𝑞_𝑗𝑄_𝑗−𝑃_𝑗,Δ=𝑃²_𝑗+1+𝑄_𝑗𝑄_𝑗+1𝛼_𝑗,=𝑃_𝑗+√Δ𝑄_𝑗𝑞_𝑗,=⌊𝛼_𝑗⌋. (2.5)(2.6)(2.7)(2.8)

Since we are primarily concerned with the case Δ≡1(mod4), we assume this for the balance of the discussion.

We need to link quadratic irrationals associated with discriminant Δ to 𝒪_Δ-ideals, namely, ideals in 𝒪_Δ√=ℤ[1+Δ2√]=ℤ⊕(1+Δ2)ℤ, (2.9) the ring of integers or maximal order in ℚ(√Δ)—see [11, Theorem  1.77, page 41]. We begin with the following.

Theorem 2.3 (ideal criterion). Let 𝐼 be a nonzero ℤ-submodule of 𝒪_Δ. Then 𝐼 has a representation in the form 𝐼=[𝑎,𝑏^′+𝑐(1+√Δ)2], (2.10) where 𝑎,𝑐∈ℕ, and 0≤𝑏^′<𝑎. Furthermore, 𝐼 is an 𝒪_Δ-ideal if and only if this representation satisfies 𝑐∣𝑎, 𝑐∣𝑏^′, and 𝑎𝑐∣𝑁((𝑏^′+𝑐(1+√Δ)/2)).

Proof. See [4, Theorem  1.2.1, page 9] or [12, Theorem  3.5.1, page 173].

Remark 2.4. If 𝑐=1, then 𝐼=[𝑎,(𝑏+√Δ)/2] is called a primitive 𝒪_Δ-ideal, where 𝑏=2𝑏^′+1 in Theorem 2.3, when 𝑏²≡Δ(mod4𝑎). Furthermore, there is a one-to-one correspondence between the primitive 𝒪_Δ-ideals and quadratic irrationals of the form 𝑃𝛼=^′+(1+√Δ)/2𝑄^′=2𝑃^′+1+√Δ2𝑄^′=𝑃+√Δ𝑄, (2.11) where 𝑃=2𝑃^′+1, 𝑄=2𝑄^′, and 𝑃²≡Δ(mod𝑄). To see this, let 𝐼=[𝑎,(𝑏+√Δ)/2] be a primitive 𝒪_Δ-ideal, and set 𝛼=(𝑏+√Δ)/(2𝑎), which is a quadratic irrational, since 𝑏²≡Δ(mod4𝑎) by Theorem 2.3. By setting 𝑃=𝑏 and 𝑄=2𝑎, then 𝛼=(𝑃+√Δ)/𝑄 and 𝐼=[𝑄,/2,(𝑃+√Δ)/2]. Thus, to each primitive 𝒪_Δ-ideal there exists a quadratic irrational of the form (2.11).
Conversely, suppose that we have a quadratic irrational of the form (2.11). Then set 𝑎=|𝑄|/2 and 𝑏=𝑃. Then 𝐼=[𝑎,(𝑏+√Δ)/2] is a primitive 𝒪_Δ-ideal by Theorem 2.3, so to each quadratic irrational of type (2.11), there corresponds a primitive 𝒪_Δ-ideal.

Example 2.5. It is possible to have a quadratic irrational of type (2.11) corresponding to a nonprimitive 𝒪_Δ-deal. However, this does not alter the fact that there is a one-to-one correspondence between them and the primitive 𝒪_Δ-deals, as demonstrated in Remark 2.4. For instance, the principal ideal 𝐼=(4)=[4,1+√5] is not primitive in 𝒪_Δ=𝒪₅=ℤ[(1+√5)/2] since 𝑐=2. Yet the quadratic irrational 𝛼=(1+√5)/4 is of type (2.11). But 𝛼 corresponds to the primitive ideal [2,(1+√5)/2] via the methodology in Remark 2.4. However, it is worthy of note that if we allow nonmaximal orders, then this permits the solution of an interesting Diophantine problem as follows. If Δ≡5(mod8) and we conisder the nonmaximal order ℤ[√Δ], then the Diophantine equation |𝑥²−Δ𝑦²|=4 with gcd(𝑥,𝑦)=1 is solvable if and only if 𝐼=[4,1+√Δ] is a principal ideal in ℤ[√Δ]—see [4, Exercise  2.1.16, page 61] and [4, Section  1.5, pages 23–30] for background details on nonmaximal orders.
Also, to see why we must specialize to quadratic irrationals of type (2.11), we have 2+√5, which is a quadratic irrational by Definition 2.1, but is not of type (2.11). Moreover, it corresponds to the ideal [1,2+√5], which is not primitive, and it does not correspond to any primitive ideal as does 𝛼 above.

Remark 2.4 and Example 2.5 motivate the following.

Definition 2.6 (ideals and quadratic irrationals). To each quadratic irrational 𝛼=(𝑃+√Δ)/𝑄, with 𝑃 odd, 𝑄 even, (and Δ≡1(mod4)), there corresponds the primitive 𝒪_Δ-ideal 𝐼=[|𝑄|2,(𝑃+√Δ)2]. (2.12) We denote this ideal by [𝛼]=𝐼 and write ℓ(𝐼) for ℓ(𝛼).

The infinite simple continued fraction of 𝛼 given by ⟨𝑞₀;𝑞₁,𝑞₂,…⟩ is called periodic (sometimes called eventually periodic), if there exists an integer 𝑘≥0 and ℓ∈ℕ such that 𝑞_𝑛=𝑞_𝑛+ℓ for all integers 𝑛≥𝑘. We use the notation 𝛼=⟨𝑞₀;𝑞₁,…,𝑞_𝑘−1,𝑞_𝑘,𝑞_𝑘+1,…,𝑞_ℓ+𝑘−1⟩, (2.13) as a convenient abbreviation. The smallest such natural number ℓ=ℓ(𝛼) is called the period length of 𝛼, and 𝑞₀,𝑞₁,…,𝑞_𝑘−1 is called the preperiod of 𝛼. If 𝑘 is the least nonnegative integer such that 𝑞_𝑛=𝑞_𝑛+ℓ for all 𝑛≥𝑘, then 𝑞_𝑘,𝑞_𝑘+1,…,𝑞_𝑘+ℓ−1 is called the fundamental period of 𝛼. In particular, we consider the so-called principal surd of 𝒪_Δ, 𝛼=(1+√Δ)/2 for which it is known that 1+√Δ2=⟨𝑞₀;𝑞₁,…,𝑞_ℓ−1,2𝑞₀−1⟩. (2.14)

We will need the following facts concerning period length.

If ℓ(𝛼)=ℓ is even, then 𝑃_ℓ/2=𝑃_ℓ/2+1, (2.15) and if ℓ is odd, then 𝑄_(ℓ+1)/2=𝑄_(ℓ−1)/2. (2.16) Furthermore, since we are assuming Δ≡1(mod4), then for 1≤𝑗≤ℓ(𝛼), 𝑄_𝑗=𝑄₀𝑄=2iff𝑗=ℓ(𝛼),_𝑗iseven𝑃_𝑗isoddforallsuch𝑗. (2.17)(2.18) Now we link pure periodicity with an important concept that will lead to the intimate link with ideals.

Definition 2.7 (reduced quadratic irrationals). Let 𝛼=(𝑃+√Δ)/𝑄 be a quadratic irrational. If 𝛼>1 and −1<𝛼^′<0, then 𝛼 is called reduced.
The next result sets the stage for our primary discussion.

Theorem 2.8 (pure periodicity equals reduction). Let 𝛼=⟨𝑞₀;𝑞₁,…⟩ be an infinite simple continued fraction, with ℓ(𝛼)=ℓ∈ℕ. Then 𝛼 is reduced if and only if 𝛼 is purely periodic, which means 𝑘=0 in (2.13), namely, 𝛼=⟨𝑞₀;𝑞₁,…,𝑞_ℓ−1⟩. (2.19)

Proof. See [10, Theorem  5.12, page 228].

Note that the notion of reduction for quadratic irrationals translates to ideals, namely we have the following.

Definition 2.9 (reduced ideals). An 𝒪_Δ-ideal is said to be reduced if it is primitive and does not contain any nonzero element 𝛼 such that both |𝛼|<𝑁(𝐼) and |𝛼^′|<𝑁(𝐼).

To see how this is tied to Definition 2.7, we need the following.

Theorem 2.10 (reduced ideals and quadratic irrationals). 𝐼=[𝑎,(𝑏+√Δ)/2] is reduced if and only if there is a 𝛽∈𝐼 such that 𝐼=[𝑁(𝐼),𝛽] with 𝛽>𝑁(𝐼) and −𝑁(𝐼)<𝛽^′<0.

Proof. See [4, Lemma  1.4.1, page 19] or [12, Theorem  5.5.1, page 258].

Corollary 2.11. If 𝐼=[𝑎,(𝑏+√Δ)/2] is a primitive 𝒪_Δ-ideal, with 𝛾=(𝑏+√Δ)/𝑎>1 and −1<(𝑏−√Δ)/2<0, then 𝐼 is reduced.

Now, we let 𝒞_Δ be the ideal-class group of 𝒪_Δ and ℎ_Δ=|𝒞_Δ| the ideal class number. If 𝐼,𝐽 are 𝒪_Δ-ideals, then equivalence of classes in 𝒞_Δ is denoted by 𝐼∼𝐽, and the class of 𝐼 is denoted by 𝐈. The following is crucial to the interplay between ideals and continued fractions, known as the infrastructure theorem for real quadratic fields or the continued fraction algorithm. (This holds for arbitrary Δ, not just Δ≡1(mod4).)

Theorem 2.12 (the continued fraction algorithm). Let 𝐼=𝐼₁=[𝑄₀/2,(𝑃₀+√Δ+1)/2] be an 𝒪_Δ-ideal corresponding to the quadratic irrational 𝛼=𝛼₀=(𝑃₀+√Δ)/2, and let 𝑃_𝑗,𝑄_𝑗 be as given in (2.5)–(2.7). If 𝐼_𝑗=[𝑄_𝑗−1/2,(𝑃_𝑗−1+√Δ)/2], then 𝐼₁∼𝐼_𝑗 for all 𝑗≥1. Moreover, there exists a least value 𝑚∈ℕ such that 𝐼_𝑚+𝑖 is reduced for all 𝑖≥0.

Proof. See [4, Theorem  2.1.2, page 44].

Corollary 2.13. A reduced 𝒪_Δ-ideal, 𝐼=[𝑄/2,(𝑃+√Δ/2] for Δ≡1(mod4) is principal if and only if 𝑄=𝑄_𝑗 for some positive integer 𝑗≤ℓ((1+√Δ)/2) in the simple continued fraction expansion of (1+√Δ)/2.

Proof. See [13].

Remark 2.14. The infrastructure given in Theorem 2.12 demonstrates that if we begin with any primitive 𝒪_Δ-ideal 𝐼, then after applying the continued fraction algorithm to 𝛼=𝛼₀, we must ultimately reach a reduced ideal 𝐼_𝑚∼𝐼 for some 𝑚≥1. Furthermore, once we have produced this ideal 𝐼_𝑚, we enter into a periodic cycle of reduced ideals, and this periodic cycle contains all the reduced ideals equivalent to 𝐼.

By Remark 2.14, once we have achieved a reduced ideal 𝐼_𝑚 via the continued fraction algorithm, then the cycle becomes periodic. Thus, it makes sense to have a name for this period length. This is given in what follows, motivated by Definition 2.6 and the continued fraction algorithm.

Definition 2.15 (cycles and periods of reduced ideals). If 𝐼=𝐼₁=[𝑄/2,(𝑃+√Δ)/2] is a reduced 𝒪_Δ-ideal and ℓ is the least positive integer such that 𝐼₁=𝐼_ℓ+1=[𝑄_ℓ/2,(𝑃_ℓ+√Δ)/2], then 𝛼_𝑗=(𝑃_𝑗+√Δ)/𝑄_𝑗 for 𝑗≥0 all have the same period length ℓ(𝛼_𝑗)=ℓ(𝛼₀)=ℓ(𝛼) via [𝛼_𝑗]=𝐼_𝑗+1𝑄=[^𝑗2,(𝑃_𝑗+√Δ)2]. (2.20) We denote this common value by ℓ=ℓ(𝒞) where 𝒞 is the equivalence class of 𝐼 in 𝒞_Δ, and call this value the period length of the cycle of reduced ideals equivalent to 𝐼. If we wish to keep track of the specific ideal, then we write ℓ(𝐼) for ℓ.

Remark 2.16. If 𝐼=[𝑄/2,(𝑃+√Δ)/2] is a reduced 𝒪_Δ-ideal, then the set {𝑄₁2,𝑄₂2𝑄,…,^ℓ2} (2.21) represents the norms of all the reduced ideals equivelnt to 𝐼 (via the continued fraction expansion of 𝛼=(𝑃+√Δ)/𝑄).

3. Prime-Producing Polynomials

We begin by stating a very palatable result by Biro that we will employ in our classification.

Lemma 3.1 (Chowla's conjecture verified). If Δ=4𝑝²+1 is square-free with some integer 𝑝>1861, then ℎ_Δ>1.

Proof. See [14, Corollary, page 179].

Corollary 3.2. The only values for which ℎ_Δ=1 with Δ=4𝑝²+1 square-free are given by 𝑝∈{1,2,3,5,7,13}.

In what follows, Δ=4𝑚+1 for 𝑚∈ℕ, and 𝑞∈ℕ is a square-free divisor of Δ, with 𝐹_Δ,𝑞(𝑥)=𝑞𝑥²+𝑞𝑥−𝑚. (3.1)

𝐹_Δ,𝑞(𝑥) is called the Euler-Rabinowitsch polynomial, which was introduced by this author in [4, Chapter 4], to discuss prime-producing quadratic polynomials and is a generalization of 𝑓_𝑚(𝑥)=𝐹_Δ,1(𝑥) used in [6, 7], where he dubbed it the Rabinowitsch polynomial. We now show how all Rabinowitsch polynomials may be determined.

Theorem 3.3. If |𝑓_𝑚(𝑥)|=|𝐹_Δ,1(𝑥)| is prime for all 𝑥∈[1,√𝑚], then Δ=4𝑝²+1 for some prime 𝑝 and ℎ_Δ=1. Also, the only values for which the above holds are Δ∈{17,37,101,197,677}. (3.2)

Proof. First we show that Δ cannot be a perfect square. If Δ=𝑟², then 𝐹_Δ,1(𝑟−12)=(𝑟−12)²+𝑟−12−𝑟²−14=0, (3.3) contradicting the hypothesis since (𝑟−1)/2<√𝑚.
Now we prove that Δ must be square-free. If Δ=𝑟²Δ₀, then Δ₀≠1 since Δ is not a perfect square. Hence, Δ₀≥5. Also, if 𝑟>√𝑚, then
𝑟²𝑟>𝑚=²Δ₀−14≥5𝑟²−14>𝑟², (3.4) a contradiction, so 𝑟≤√𝑚. Therefore, since 𝐹_Δ,1(𝑟−12)=𝑟²(1−Δ₀4), (3.5) this contradicts the hypothesis if 𝑟>1. Hence, 𝑟=1, and Δ is square-free and so may be used for simple continued fraction expansions in the maximal order 𝒪_Δ=ℤ[(1+√Δ)/2].
If 𝑚 is even, then
|𝐹_Δ,1𝑚(2)|=|2(3−2)| (3.6) is composite unless 𝑚=4, namely, unless Δ=17, observing that 𝑚≠2,6 since 4⋅2+1=3² and 4⋅6=1=5². Thus, we may assume that 𝑚 is odd.
In the continued fraction expansion of 𝛼=(1+√Δ)/2, Δ=𝑃²_𝑗+𝑄_𝑗𝑄_𝑗−1 for all natural numbers 𝑗≤ℓ(𝛼) by (2.6). We now show that ℓ(𝛼)=3.
Suppose that ℓ(𝛼)>3. By (2.18), for each 𝑗=1,2,…,ℓ(𝛼), we may set
𝑥_𝑗=𝑃_𝑗−12. (3.7) Since 1≤𝑃_𝑗<√Δ by (2.6), then 0≤𝑥_𝑗≤(√Δ−1)/2. If 𝑄_𝑗=2 for any 𝑗=1,2,3, then by (2.17), ℓ(𝛼)≤3, a contradiction. Thus, 𝑄_𝑗≠2 for 𝑗∈{1,2,3}. However, if 𝑥_𝑗≠0, then |𝐹_Δ,1(𝑥_𝑗)| is prime by hypothesis since 𝑥_𝑗≤⌊√𝑚⌋ for 𝑗=1,2,3. We have, by (2.6), that |𝐹_Δ,1(𝑥_𝑗)|=|𝑥²_𝑗+𝑥_𝑗𝑃−𝑚|=|^2𝑗−Δ4𝑄|=^𝑗𝑄_𝑗−14. (3.8) Therefore, 𝑄_𝑗𝑄_𝑗−1=4𝑝 where 𝑝 is prime if 𝑥_𝑗≠0. Now suppose that 𝑗=2. Since 𝑄_𝑗 is even for all 𝑗∈ℕ by (2.18), then 𝑄₁=2𝑝 and 𝑄₂=2, a contradiction as above. We have shown that 𝑥₂=0. If 𝑗=3, then by the same argument 𝑄₃=2, a contradiction. We have shown that 𝑥₃=0. Hence, by (2.5), 𝑃₃=1=𝑞₂𝑄₂−𝑃₂=𝑞₂𝑄₂−1, (3.9) which implies that 2=𝑞₂𝑄₂ forcing 𝑄₂=2, a contradiction. We have shown that ℓ(𝛼)≤3.
If ℓ(𝛼)=1, then 𝑄₀=𝑄₁=2, and Δ=𝑃²₁+4 with
|𝐹_Δ,1(𝑥₁𝑄)|=¹𝑄₀4=1, (3.10) contradicting the hypothesis unless 𝑥₁=0 which means Δ=5. However, 𝐹_5,1(1)=1, where 1∈[1,√𝑚]={1}, contradicting the hypothesis. Hence, ℓ(𝛼)>1.
If ℓ(𝛼)=2, then by (3.8), 𝑄₁=2𝑝 for a prime 𝑝, 𝑄₂=2, and 𝑃₂=𝑃₁ by (2.15). Hence, by (2.5)
𝑃₁=𝑃₂=𝑞₁2𝑝−𝑃₁, (3.11) which implies that 𝑃₁=𝑞₁𝑝. Now we show that 𝑞₁=1.
Assume to the contrary that 𝑞₁≥3, since 𝑃₁ is odd by (2.18). Then by (2.6), Δ=𝑃²₁+4𝑝. Therefore 𝑝<√Δ−2. If we let 𝑥=(𝑝−1)/2≤√𝑚, then
|𝐹_Δ,1𝑝(𝑥)|=|²−Δ4|=𝑝|𝑝−Δ/𝑝4|=𝑝, (3.12) by hypothesis, which forces 𝑝−Δ/𝑝=−4, namely, Δ=𝑝²+4𝑝, and 𝑞₁=1. Therefore, 𝑚=(𝑝²+4𝑝−1)/4, so (𝑝+1)/2<√𝑚<(𝑝+2)/2, which implies that ⌊√𝑚⌋=(𝑝+1)/2. Therefore, 𝐹_Δ,1(𝑝+12)=(𝑝+12)²+𝑝+12𝑝−(²+4𝑝−14)=1, (3.13) contradicting the hypothesis which says 𝐹_Δ,1(𝑥) is prime for all 𝑥∈[1,√𝑚]. We have shown that ℓ(𝛼)=3. Thus, 𝑄₂≠2, so by (3.8), for 𝑗=2, we get 𝑥₂=0, which means 𝑃₂=1. For 𝑗=1, (3.8) tells us that 𝑄₁=2𝑝 where 𝑝 is prime since 𝑄₀=2. By (2.16), 𝑄₂=𝑄₁ since ℓ(𝛼)=3, so by (2.6) Δ=4𝑚+1=𝑃²₂+𝑄₂𝑄₁=1+4𝑝², (3.14) as we sought to show. Now we show that ℎ_Δ=1 for these values.
By Theorem 2.12, if 𝐈 is an ideal class in 𝒞_Δ, then 𝐈 contains a reduced ideal 𝐼=[𝑄/2,(𝑃+√Δ)/2]. Using a similar argument to the above on 𝛽=(𝑃+√Δ)/𝑄 as we did for 𝛼=(1+√Δ)/2, we achieve that 𝐼 is in a cycle of period length 3, namely ℓ(𝛽)=ℓ(𝐼)=3. Now in the simple continued fraction expansion of 𝛽, let 𝑄=𝑄₀ and 𝑃=𝑃₀. Then, as in the case for 𝛼, (where we use the same symbols 𝑄_𝑗 without risk of confusion since we are done with 𝛼), 𝑄₂=𝑄₁, 𝑃₂=1, and by (3.8) applied to 𝛽s values of 𝑄_𝑗, we must have that 𝑄₁𝑄₀=4𝑞 for some prime 𝑞. If 𝑄₀=2, then by Corollary 2.13, 𝐼∼1. If 𝑄₀>2, then since 𝑄₁ is even by (2.18), we must have either 𝑞=2=𝑄₁ and 𝑄₀=4, or 𝑄₀=2𝑞 and 𝑄₁=2. In either case, by Corollary 2.13 again, 𝐼∼1. Hence, ℎ_Δ=1.
By Lemma 3.1, the only values for which the result holds are in the list (3.2).

The following is the affirmative solution of four conjectures by this author posed in 1988 in [8, Conjectures  1–4, page 20]—see also [15, page 311]. Note that the equivalence of the conjectures follows from [2].

Corollary 3.4. For a prime 𝑝=4𝑚+1=4𝑞²+1, where 𝑞 is prime, |𝑓_𝑞2(𝑥)|=|𝑥²+𝑥−𝑞²| is prime for 𝑥∈[1,𝑞] if and only if 𝑞≤13.

Corollary 3.5. Suppose that 𝑝=4𝑞²+1 is prime, where 𝑞 is prime. Then all odd primes 𝑟<𝑞 are inert in ℚ(√𝑝) if and only if 𝑞≤13.

Corollary 3.6. Suppose that 𝑝=4𝑞²+1 is prime, where 𝑞 is prime. Then 𝑓_𝑞2(𝑥)≢0(mod𝑟) for all positive integers 𝑥 and primes 𝑟 satisfying 𝑥<𝑟<𝑞 if and only if 𝑞≤13.

Corollary 3.7. Suppose that 𝑝=4𝑞²+1 is prime, where 𝑞 is prime and 𝐹=ℚ(√𝑝) with Dedekind-zeta function 𝜁_𝐹. Then 2𝜁_𝐹(−1)=𝑞(2𝑞²+7)/45 if and only if 𝑞≤13.

Example 3.8. A nice illustration of Corollary 3.7 is for 𝑞=3, with 𝑝=37, where 𝜁_𝐹(−1)=𝑞(2𝑞²+7)=5906. (3.15) Now we look at a slight variation that captures more of the results in [6, 7], as well as some missed by them. We will be using the following other beautiful result by Biro.

Lemma 3.9 (Yokoi's conjecture verified). If Δ=𝑝²+4 is square-free for some odd integer 𝑝>1861, then ℎ_Δ>1.

Proof. See [16].

Corollary 3.10. If ℎ_Δ=1 for Δ=𝑝²+4 square-free, then 𝑝∈{1,3,5,7,13,17}.

As well, we will be employing the following equally pleasant result by Byeon, Kim, and Lee, who used methods similar to those of Biro.

Lemma 3.11 (Mollin's conjecture verified). If Δ=𝑛²−4 is square-free, then ℎ_Δ>1 for 𝑛>21.

Proof. See [9].

Corollary 3.12. If ℎ_Δ=1 for Δ=𝑛²−4 square-free, then 𝑛∈{3,5,9,21}.

Theorem 3.13. If |𝑓_𝑚(𝑥)|=|𝐹_Δ,1(𝑥)| is 1 or prime for all 𝑥∈[0,√𝑚−1], where Δ=4𝑚+1, then for Δ≠9, either Δ=𝑛²−4forsome𝑛∈ℕ,ℎ_Δ=1, (3.16) or Δ=𝑝²+4foraprime𝑝>2,ℎ_Δ=1. (3.17) Also, the only values for which (3.16) holds are Δ∈{5,21,77,437}, (3.18) and the only values for which (3.17) holds are Δ∈{13,29,53,173,293}. (3.19)

Proof. If Δ=𝑟², then by hypothesis |𝐹_Δ,1(0)|=(𝑟²−1)/4=𝑝, where 𝑝 is prime. Thus, (𝑟−1)(𝑟+1)=4𝑝, from which we deduce that the only possibility is 𝑝=2=𝑚 and 𝑟=3, namely, Δ=9, contradicting the hypothesis. Thus, Δ is not a square. Moreover, by the same argument as in the proof of Theorem 3.3, Δ is square-free. Hence, we may apply continued fraction theory as above.
If 𝑚 is even, then 𝐹_Δ,1(0)=−𝑚, contradicting the hypothesis unless 𝑚=2, for which Δ=9. Hence we may assume that 𝑚 is odd and since 𝑚=1 gives Δ=5 which satisfies the hypothesis, we assume that 𝑚>1 is odd.
Let 𝛼=(1+√Δ)/2, so in the continued fraction expansion of 𝛼, Δ=𝑃²_𝑗+𝑄_𝑗𝑄_𝑗−1 for 1≤𝑗≤ℓ(𝛼) by (2.6). If 𝑥_𝑗 is given by (3.7), then by (3.8), we see that since 𝑥_𝑗∈[0,√𝑚−1], then by hypothesis
|𝐹_Δ,1(𝑥_𝑗𝑄)|=^𝑗𝑄_𝑗−14 (3.20) is prime for 𝑗=1,2,…,ℓ(𝛼). In particular, 𝑄₁=2𝑝 for a prime 𝑝, and 𝑄₁𝑄₂=4𝑞 for a prime 𝑞. However, since 𝑄_𝑗 is even for all 𝑗 by (2.18), then 𝑄₂=2, and 𝑄₁=2𝑞 is the only possibility. Thus, ℓ(𝛼)=2, so 𝑝=𝑞 and 𝑄₁=𝑄₂. By the same argument as in the proof of Theorem 3.3, Δ=𝑝²+4𝑝=(𝑝+2)²−4.
By virtually the same argument as used in the proof of Theorem 3.3, we get ℎ_Δ=1. However, by Corollary 3.12, the values of Δ are those in the list (3.18).
Lastly, we may assume that ℓ(𝛼)=1, namely, Δ=𝑃²₁+4. Again, by the same argument as used in the proof of Theorem 3.3, we get that 𝑃₁=𝑝, a prime, and ℎ_Δ=1. Thus, by Corollary 3.10, the values are those in the list (3.19).

Putting Theorems 3.3 and 3.13 together, we get an (unconditional) update on the Rabinowitsch-Mollin-Williams Theorem as follows. This is a complete determination of all narrow Richaud-Degert types with class number 1, for which there exist exactly 14 Rabinowitsch polynomials, based upon the recent solution of the Chowla, Mollin, and Yokoi conjectures in Lemmas 3.1–3.11. Note as well that in both [7, 17] it is proved there are only finitely many Rabinowitsch polynomials 𝑓_𝑚.

We list the 14 values (of narrow Richaud-Degert types) unconditionally in Theorem 3.14, whereas the remaining list of four wide Richaud-Degert types is complete with one possible exception, whose existence would be a counterexample to the GRH. We list the 18 Rabinowitsch polynomials below, excluding the degenerate case of Δ=9 which is included in the 14 values in [7].

Note, as well, that although the original Theorem 1.1 only considers the values of 𝑥∈[1,√𝑚], and Theorem 3.13 considers 𝑥∈[0,√𝑚−1], the value of the Rabinowitsch polynomials therein also has |𝑓_𝑚(⌊√𝑚⌋)| being 1 or prime as well. The restriction in Theorem 3.13 for the range of 𝑥 values was made to be in synch with the setup in [6, 7] in order to correct and complete their results. Hence, the following is indeed an update and an unconditional rendering of the original.

Theorem 3.14 (Rabinowitsch-Mollin-Williams updated). If Δ=4𝑚+1, 𝑚≠2, then the followings are equivalent. (a)|𝑓_𝑚(𝑥)|=|𝑥²+𝑥−𝑚| is 1 or prime for all 𝑥∈[1,√𝑚].(b)ℎ_Δ=1 and Δ is one of the following forms:(i)𝑛²−4 for some 𝑛∈ℕ,(ii)𝑝²+4 for a prime 𝑝>2,(iii)4𝑝²+1 for a prime 𝑝.(c)Δ∈{5,13,17,21,29,37,53,77,101,173,197,293,437,677}.

Remark 3.15. This remark is provided for the sake of completeness and explaining details in extending the results in [7]. Therein the authors missed all of the values 21,77, and 437. The value 21 is of their type (iii) with, in their notation, 𝑥₀=0, 𝑛=−1, 𝑡=2, and 𝑚=5, so the corresponding Rabinowitsch polynomial is 𝑓₅(𝑥)=𝑥²+𝑥−5whichisprimefor𝑥∈[0,1]=[𝑥₀,𝑥₀+𝑡−1]. (3.21) The value 77 is of type (iii) with 𝑥₀=0, 𝑛=−1, 𝑡=4, and 𝑚=19, with Rabinowitsch polynomial 𝑓₁₉(𝑥)=𝑥²+𝑥−19beingprimefor𝑥∈[0,3]=[𝑥₀,𝑥₀+𝑡−1]. (3.22) Indeed, 𝑓₁₉(𝑥) is prime or 1 for all 𝑥∈[0,9] or three times the length. Lastly, 437 is of type (iii) with 𝑥₀=0, 𝑛=−1, 𝑡=10, and 𝑚=109, with 𝑓₁₀₉(𝑥)=𝑥²+𝑥−109whichis1orprimefor𝑥∈[0,9]=[𝑥₀,𝑥₀+𝑡−1]. (3.23) Again, here 𝑓₁₀₉(𝑥) is 1 or prime for triple the length, namely, for 𝑥∈[0,27]. These 14 values are exactly the values listed in [4, Table  4.2.3, page 139], after the statement of the Rabinowitsch-Mollin-Williams Theorem therein. Also in the following we capture the remaining values from [7] and others they missed.

The following deals with wide Richaud-Degert types and captures the balance of the values using the Euler-Rabinowitsch polynomial 𝐹_Δ,𝑝(𝑥) for a prime 𝑝 dividing Δ. Recall that ℎ_Δ=1 for a composite Δ can occur only if Δ=𝑝𝑞, where 𝑝≡𝑞≡3(mod4), are primes.

In [4, Conjecture  4.2.1, page 140], we provided the following conjecture for wide Richaud-Degert types that remains open.

Conjecture. If Δ=𝑝𝑞≡5(mod8), where 𝑝≡𝑞≡3(mod4), are primes with 𝑝<𝑞, then the following are equivalent.(a)|𝐹_Δ,𝑝(𝑥)| is 1 or prime for all 𝑥∈[0,(√Δ−2)/4].(b)Δ=𝑝²𝑠²±4𝑝 of Δ=4𝑝²𝑠²−𝑝 for some 𝑠∈ℕ and ℎ_Δ=1.

Remark 3.16. We have a list of values for Conjecture 1, which as above, we know is valid with one possible GRH-ruled-out exception. It is Δ∈{33,69,93,141,213,237,413,453,573,717,1077,1133,1253,1293,1757}. (3.24) The use of 𝐹_Δ,𝑝(𝑥) is much less demanding than the use of 𝑓_𝑚(𝑥), and the lone two values found in [7] attest to this. However, they missed two other values for 𝑓_𝑚(𝑥) that we now provide and we are able to pose a new conjecture on the basis of it, which does not appear in literature thus far.

Conjecture 2. If 1+4𝑚=Δ=𝑝𝑞 with 𝑝<𝑞 primes and |𝑓_𝑚(𝑥)| is prime for all 𝑥∈[(𝑝+1)/2,√𝑚+(𝑝−1)/2], then Δ=9𝑝²±4𝑝foranoddprime𝑝,ℎ_Δ=1. (3.25) Moreover, the only values for which (3.25) holds are Δ∈{69,93,413,1133}. (3.26)

By the above discussion, we know that the list (see Table 1) in (3.26) is complete with one possible GRH-ruled-out exception. The wide Richaud-Degert values missed in [7] are Δ=69 and Δ=93. We now have a complete list of the 18 Rabinowisch polynomials with one possible exception on the wide Richaud-Degert types, where we exclude Δ=9 for reasons given above. If we included the latter then the corrected list in [7] grows from 14 to 19 values.

Table 1 

Remark 3.17. After the writing of this paper Anitha Srinivasan informed me that, in an unpublished manuscript, she has proved Conjecture 2. Thus, we will address this and other matters in later joint work.

Acknowledgments

The author gratefully acknowledges the support of NSERC Canada Grant no. A8484. Moreover, thanks go to the referee for suggestions that led to the clarification, increased readability, and streamlining of the presentation.