IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation81906810.1155/2009/819068819068Research ArticleThe Rabinowitsch-Mollin-Williams Theorem RevisitedMollinR. A.KriegAloysDepartment of Mathematics and StatisticsUniversity of CalgaryCalgary, ABCanadaT2N 1N4ucalgary.ca20092408200920092404200924062009200820092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We completely classify all polynomials of type (x2+x(Δ1))/4 which are prime or 1 for a range of consecutive integers x0, called Rabinowitsch polynomials, where Δ1(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 , says that if Δ=1-4m is square-free, then the class number, hΔ, of the complex quadratic field (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  by this author and Williams. In  and in subsequent renderings of the result, we considered all values of Δ. However, the case where Δ1(mod4) is essentially trivial, and the values (unconditionally) known for these Rabinowitsch polynomials are Δ{2,3,6,7,11}—see . Therefore, we consider only the interesting case, namely, Δ1(mod4).

Theorem 1.1 (Rabinowitsch-Mollin-Williams).

If Δ=1+4m, where m, then the following are equivalent.

fm(x)=x2+x-m is 1 or prime for all integers x[1,m].

hΔ=1 and Δ=s2+r, where r{1,±4}.

Proof.

See , 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  in 2002. Then in 2003 , 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 . 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 r4s. 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 . (Here GRH means the generalized Riemann hypothesis.) Lastly, we show how four conjectures posed by this author in 1988 in  are affirmatively settled via the above and complement another conjecture by this author affirmatively verified by Byeon et al. in .

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  or for a more advanced approach in .

We denote the infinite simple continued fraction expansion of a given α by α=q0;q1,q2,  where  qj  for  j,q0=α, 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.

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 and a0.

Remark 2.2.

A real number α is a quadratic irrational if and only if there exist P,Q,Δ such that Q0, Δ is not a perfect square, and α=P+ΔQ,(P,Q). Moreover, if α is a quadratic irrational, then Q(P2-Δ). Also, α=(P-Δ)Q is called the algebraic conjugate of α. Here both α and α are the roots of f(x)=x2-Tr(α)x+N(α), 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 j0 define Pj+1=qjQj-Pj,Δ=Pj+12+QjQj+1,  αj=Pj+ΔQj,qj=αj.

Since we are primarily concerned with the case Δ1(mod4), we assume this for the balance of the discussion.

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

Theorem 2.3 (ideal criterion).

Let I be a nonzero -submodule of 𝒪Δ. Then I has a representation in the form I=[a,b+c(1+Δ)2], where a,c, and 0b<a. Furthermore, I is an 𝒪Δ-ideal if and only if this representation satisfies ca, cb, and acN((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 𝒪Δ-ideal, where b=2b+1 in Theorem 2.3, when b2Δ(mod4a). Furthermore, there is a one-to-one correspondence between the primitive 𝒪Δ-ideals and quadratic irrationals of the form α=P+(1+Δ)/2Q=2P+1+Δ2Q=P+ΔQ, where P=2P+1, Q=2Q, and P2Δ(modQ). To see this, let I=[a,(b+Δ)/2] be a primitive 𝒪Δ-ideal, and set α=(b+Δ)/(2a), which is a quadratic irrational, since b2Δ(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 𝒪Δ-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 𝒪Δ-ideal by Theorem 2.3, so to each quadratic irrational of type (2.11), there corresponds a primitive 𝒪Δ-ideal.

Example 2.5.

It is possible to have a quadratic irrational of type (2.11) corresponding to a nonprimitive 𝒪Δ-deal. However, this does not alter the fact that there is a one-to-one correspondence between them and the primitive 𝒪Δ-deals, as demonstrated in Remark 2.4. For instance, the principal ideal I=(4)=[4,1+5] is not primitive in 𝒪Δ=𝒪5=[(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 Δ5(mod8) and we conisder the nonmaximal order [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 [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 Δ1(mod4)), there corresponds the primitive 𝒪Δ-ideal I=[|Q|2,(P+Δ)2]. We denote this ideal by [α]=I and write (I) for (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 k0 and such that qn=qn+ for all integers nk. We use the notation α=q0;q1,,qk-1,qk,qk+1,,q+k-1¯, as a convenient abbreviation. The smallest such natural number =(α) is called the period length of α, and q0,q1,,qk-1 is called the preperiod of α. If k is the least nonnegative integer such that qn=qn+ for all nk, then qk,qk+1,,qk+-1 is called the fundamental period of α. In particular, we consider the so-called principal surd of 𝒪Δ, α=(1+Δ)/2 for which it is known that 1+Δ2=q0;q1,,q-1,2q0-1¯.

We will need the following facts concerning period length.

If (α)= is even, then P/2=P/2+1, and if is odd, then Q(+1)/2=Q(-1)/2. Furthermore, since we are assuming Δ1(mod4), then for 1j(α), Qj=Q0=2iff  j=(α),Qjis  evenPj  is  oddfor  all  such  j.   Now we link pure periodicity with an important concept that will lead to the intimate link with ideals.

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 (α)=. Then α is reduced if and only if α is purely periodic, which means k=0 in (2.13), namely, α=q0;q1,,q-1¯.

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 𝒪Δ-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 βI 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 𝒪Δ-ideal, with γ=(b+Δ)/a>1 and -1<(b-Δ)/2<0, then I is reduced.

Now, we let 𝒞Δ be the ideal-class group of 𝒪Δ and hΔ=|𝒞Δ| the ideal class number. If I,J are 𝒪Δ-ideals, then equivalence of classes in 𝒞Δ 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 Δ1(mod4).)

Theorem 2.12 (the continued fraction algorithm).

Let I=I1=[Q0/2,(P0+Δ+1)/2] be an 𝒪Δ-ideal corresponding to the quadratic irrational α=α0=(P0+Δ)/2, and let Pj,  Qj be as given in (2.5)–(2.7). If Ij=[Qj-1/2,(Pj-1+Δ)/2], then I1~Ij for all j1. Moreover, there exists a least value m such that Im+i is reduced for all i0.

Proof.

See [4, Theorem  2.1.2, page 44].

Corollary 2.13.

A reduced 𝒪Δ-ideal, I=[Q/2,(P+Δ/2] for Δ1(mod4) is principal if and only if Q=Qj for some positive integer j((1+Δ)/2) in the simple continued fraction expansion of (1+Δ)/2.

Proof.

See .

Remark 2.14.

The infrastructure given in Theorem 2.12 demonstrates that if we begin with any primitive 𝒪Δ-ideal I, then after applying the continued fraction algorithm to α=α0, we must ultimately reach a reduced ideal Im~I for some m1. Furthermore, once we have produced this ideal Im, 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 Im 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 𝒪Δ-ideal and is the least positive integer such that I1=I+1=[Q/2,(P+Δ)/2], then αj=(Pj+Δ)/Qj for j0 all have the same period length (αj)=(α0)=(α) via [αj]=Ij+1=[Qj2,(Pj+Δ)2]. We denote this common value by =(𝒞) where 𝒞 is the equivalence class of I in 𝒞Δ, 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 (I) for .

Remark 2.16.

If I=[Q/2,(P+Δ)/2] is a reduced 𝒪Δ-ideal, then the set {Q12,Q22,,Q2} 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 hΔ>1.

Proof.

See [14, Corollary, page 179].

Corollary 3.2.

The only values for which hΔ=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, and q is a square-free divisor of Δ, with FΔ,q(x)=qx2+qx-m.

FΔ,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 fm(x)=FΔ,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 |fm(x)|=|FΔ,1(x)| is prime for all x[1,m], then Δ=4p2+1 for some prime p and hΔ=1. Also, the only values for which the above holds are Δ{17,37,101,197,677}.

Proof.

First we show that Δ cannot be a perfect square. If Δ=r2, then FΔ,1(r-12)=(r-12)2+r-12-r2-14=0, contradicting the hypothesis since (r-1)/2<m.

Now we prove that Δ must be square-free. If Δ=r2Δ0, then Δ01 since Δ is not a perfect square. Hence, Δ05. Also, if r>m, then

r2>m=r2Δ0-145r2-14>r2, a contradiction, so rm. Therefore, since FΔ,1(r-12)=r2(1-Δ04), 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 𝒪Δ=[(1+Δ)/2].

If m is even, then

|FΔ,1(2)|=|2(3-m2)| is composite unless m=4, namely, unless Δ=17, observing that m2,6 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=Pj2+QjQj-1 for all natural numbers j(α) by (2.6). We now show that (α)=3.

Suppose that (α)>3. By (2.18), for each j=1,2,,(α), we may set

xj=Pj-12. Since 1Pj<Δ by (2.6), then 0xj(Δ-1)/2. If Qj=2 for any j=1,2,3, then by (2.17), (α)3, a contradiction. Thus, Qj2 for j{1,2,3}. However, if xj0, then |FΔ,1(xj)| is prime by hypothesis since xjm for j=1,2,3. We have, by (2.6), that |FΔ,1(xj)|=|xj2+xj-m|=|Pj2-Δ4|=QjQj-14. Therefore, QjQj-1=4p where p is prime if xj0. Now suppose that j=2. Since Qj is even for all j 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, which implies that 2=q2Q2 forcing Q2=2, a contradiction. We have shown that (α)3.

If (α)=1, then Q0=Q1=2, and Δ=P12+4 with

|FΔ,1(x1)|=Q1Q04=1, contradicting the hypothesis unless x1=0 which means Δ=5. However, F5,1(1)=1, where 1[1,m]={1}, contradicting the hypothesis. Hence, (α)>1.

If (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, which implies that P1=q1p. Now we show that q1=1.

Assume to the contrary that q13, since P1 is odd by (2.18). Then by (2.6), Δ=P12+4p. Therefore p<Δ-2. If we let x=(p-1)/2m, then

|FΔ,1(x)|=|p2-Δ4|=p|p-Δ/p4|=p, 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, FΔ,1(p+12)=(p+12)2+p+12-(p2+4p-14)=1, contradicting the hypothesis which says FΔ,1(x) is prime for all x[1,m]. We have shown that (α)=3. Thus, Q22, 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 (α)=3, so by (2.6) Δ=4m+1=P22+Q2Q1=1+4p2, as we sought to show. Now we show that hΔ=1 for these values.

By Theorem 2.12, if I is an ideal class in 𝒞Δ, 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 (β)=(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 Qj without risk of confusion since we are done with α), Q2=Q1, P2=1, and by (3.8) applied to βs values of Qj, 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, hΔ=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 .

Corollary 3.4.

For a prime p=4m+1=4q2+1, where q is prime, |fq2(x)|=|x2+x-q2| is prime for x[1,q] if and only if q13.

Corollary 3.5.

Suppose that p=4q2+1 is prime, where q is prime. Then all odd primes r<q are inert in (p) if and only if q13.

Corollary 3.6.

Suppose that p=4q2+1 is prime, where q is prime. Then fq2(x)0(modr) for all positive integers x and primes r satisfying x<r<q if and only if q13.

Corollary 3.7.

Suppose that p=4q2+1 is prime, where q is prime and F=(p) with Dedekind-zeta function ζF. Then 2ζF(-1)=q(2q2+7)/45 if and only if q13.

Example 3.8.

A nice illustration of Corollary 3.7 is for q=3, with p=37, where ζF(-1)=q(2q2+7)90=56. 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 hΔ>1.

Proof.

See .

Corollary 3.10.

If hΔ=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 hΔ>1 for n>21.

Proof.

See .

Corollary 3.12.

If hΔ=1 for Δ=n2-4 square-free, then n{3,5,9,21}.

Theorem 3.13.

If |fm(x)|=|FΔ,1(x)| is 1 or prime for all x[0,m-1], where Δ=4m+1, then for Δ9, either Δ=n2-4for  some  n,hΔ=1, or Δ=p2+4for  a  prime  p>2,hΔ=1. Also, the only values for which (3.16) holds are Δ{5,21,77,437}, and the only values for which (3.17) holds are Δ{13,29,53,173,293}.

Proof.

If Δ=r2, then by hypothesis |FΔ,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 FΔ,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=Pj2+QjQj-1 for 1j(α) by (2.6). If xj is given by (3.7), then by (3.8), we see that since xj[0,m-1], then by hypothesis

|FΔ,1(xj)|=QjQj-14 is prime for j=1,2,,(α). In particular, Q1=2p for a prime p, and Q1Q2=4q for a prime q. However, since Qj is even for all j by (2.18), then Q2=2, and Q1=2q is the only possibility. Thus, (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 hΔ=1. However, by Corollary 3.12, the values of Δ are those in the list (3.18).

Lastly, we may assume that (α)=1, namely, Δ=P12+4. Again, by the same argument as used in the proof of Theorem 3.3, we get that P1=p, a prime, and hΔ=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 3.13.11. Note as well that in both [7, 17] it is proved there are only finitely many Rabinowitsch polynomials fm.

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 .

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 |fm(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, m2, then the followings are equivalent.

|fm(x)|=|x2+x-m| is 1 or prime for all x[1,m].

hΔ=1 and Δ is one of the following forms:

n2-4 for some n,

p2+4 for a prime p>2,

4p2+1 for a prime p.

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 . 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-5which  is  prime  for  x[0,1]=[x0,x0+t-1]. The value 77 is of type (iii) with x0=0,   n=-1,   t=4, and m=19, with Rabinowitsch polynomial f19(x)=x2+x-19being  prime  for  x[0,3]=[x0,x0+t-1]. 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-109which  is  1  or  prime  for  x[0,9]=[x0,x0+t-1]. 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  and others they missed.

The following deals with wide Richaud-Degert types and captures the balance of the values using the Euler-Rabinowitsch polynomial FΔ,p(x) for a prime p dividing Δ. Recall that hΔ=1 for a composite Δ can occur only if Δ=pq, where pq3(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=pq5(mod8), where pq3(mod4), are primes with p<q, then the following are equivalent.

|FΔ,p(x)| is 1 or prime for all x[0,(Δ-2)/4].

Δ=p2s2±4p of Δ=4p2s2-p for some s and hΔ=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}. The use of FΔ,p(x) is much less demanding than the use of fm(x), and the lone two values found in  attest to this. However, they missed two other values for fm(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 |fm(x)| is prime for all x[(p+1)/2,m+(p-1)/2], then Δ=9p2±4pfor  an  odd  prime  p,  hΔ=1. Moreover, the only values for which (3.25) holds are Δ{69,93,413,1133}.

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  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  grows from 14 to 19 values.

Δm[x0,y0]  Values    of  |fm(x)|  for  x[x0,y0]
51[0,1]1,1
133[0,1]3,1
174[1,2]2,2
215[0,2]5,3,1
297[0,2]7,5,1
379[1,3]7,3,3
5313[0,3]13,11,7,1
6917[2,5]17,5,3,13
7719[0,4]19,17,13,7,1
9323[2,5]17,11,3,7
10125[1,5]23,19,13,5,5
17343[0,6]43,41,37,31,23,13,1
19749[1,7]47,43,37,29,19,7,7
29373[0,8]73,71,67,61,53,43,31,17,1
413103[4,13]83,73,61,47,31,13,7,29,53,79
437109[0,10]109,107,103,97,89,79,67,53,37,19,1
677169[1,13]167,163,157,149,139,127,113,97,79,59,37,13,13
1133283[6,21]241,227,211,193,173,151,127,101,73,43,11,
23,59,97,137,179
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.

RabinowitschG.Eindeutigkeit der zerlegung in primzahlfaktoren in quadratischen ZahlkörpernJournal für die Reine und Angewandte Mathematik1913142153164MollinR. A.WilliamsH. C.On prime valued polynomials and class numbers of real quadratic fieldsNagoya Mathematical Journal1988112143151MR974269ZBL0629.12004MollinR. A.WilliamsH. C.Prime producing quadratic polynomials and real quadratic fields of class number oneThéorie des Nombres (Quebec, PQ, 1987)1989Berlin, Germanyde Gruyter654663MR1024594ZBL0695.12002MollinR. A.Quadratics1996Boca Raton, Fla, USACRC Pressxx+387CRC Press Series on Discrete Mathematics and Its ApplicationsMR1383823MollinR. A.An elementary proof of the Rabinowitsch-Mollin-Williams criterion for real quadratic fieldsJournal of Mathematical Sciences1996711727MR1433215ByeonD.dhbyeon@math.snu.ac.krStarkH. M.hmstark@ucsd.eduOn the finiteness of certain Rabinowitsch polynomialsJournal of Number Theory2002941219221ByeonD.StarkH. M.On the finiteness of certain Rabinowitsch polynomials. IIJournal of Number Theory2003991177180MR195725310.1016/S0022-314X(02)00063-XZBL1033.11010MollinR. A.Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. ChowlaProceedings of the American Mathematical Society198810211721MR91570710.2307/2046022ZBL0673.12005ByeonD.KimM.LeeJ.Mollin's conjectureActa Arithmetica2007126299114MR228941010.4064/aa126-2-1ZBL1125.11059MollinR. A.Fundamental Number Theory with Applications20082ndBoca Raton, Fla, USAChapman & Hall/CRC, Taylor and Francisx+369Discrete Mathematics and Its ApplicationsMR2404578MollinR. A.Algebraic Number Theory1999Boca Raton, Fla, USAChapman & Hall/CRC, Taylor and Francisxiv+483Discrete Mathematics and Its ApplicationsMR1682930MollinR. A.Fundamental Number Theory with Applications19981stBoca Raton, Fla, USAChapman & Hall/CRC, Taylor and FrancisLouboutinS.MollinR. A.WilliamsH. C.Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing quadratic polynomials and quadratic residue coversCanadian Journal of Mathematics1992444824842MR1178571ZBL0771.11039BiróA.Chowla's conjectureActa Arithmetica20031072179194MR197082210.4064/aa107-2-5ZBL1154.11339RibenboimP.The Book of Prime Number Records1988New York, NY, USASpringerxxiv+476MR931080BiróA.Yokoi's conjectureActa Arithmetica2003106185104MR195697710.4064/aa106-1-6ZBL1154.11338Schlage-PuchtaJ.-C.Finiteness of a class of Rabinowitsch polynomialsArchivum Mathematicum2004403259261MR2107020ZBL1122.11070