#### Abstract

We completely classify all polynomials of type which are prime or 1 for a range of consecutive integers , called *Rabinowitsch polynomials*, where with 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 is square-free, then the class number, , of the complex quadratic field is exactly when is prime for all integers . 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 . However, the case where is essentially trivial, and the values (unconditionally) known for these Rabinowitsch polynomials are —see [3]. Therefore, we consider only the interesting case, namely, .

Theorem 1.1 (Rabinowitsch-Mollin-Williams). *If where , then the following are equivalent. *(a)* is or prime for all integers . *(b)* and where .*

*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 where , 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 where . 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 by
where is the *floor* of , namely, the greatest integer less than or equal to . 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 is called a *quadratic irrational* if it is an irrational number which is the root of where and .

*Remark 2.2. *A real number is a *quadratic irrational* if and only if there exist such that , is not a perfect square, and
Moreover, if is a quadratic irrational, then . Also,
is called the *algebraic conjugate* of . Here both and are the roots of
where is the *trace* of and is the *norm* of —see [10, Theorem 5.9, page 222].

Now, given a quadratic irrational , set , , and for define

Since we are primarily concerned with the case , *we assume this for the balance of the discussion*.

We need to link quadratic irrationals associated with discriminant to -ideals, namely, ideals in
the *ring of integers* or *maximal order* in —see [11, Theorem 1.77, page 41]. We begin with the following.

Theorem 2.3 (ideal criterion). *Let be a nonzero -submodule of . Then has a representation in the form
**
where and . Furthermore, is an -ideal if and only if this representation satisfies , , and .*

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

*Remark 2.4. *If , then is called a *primitive *-ideal, where in Theorem 2.3, when . Furthermore, there is a one-to-one correspondence between the *primitive *-ideals and quadratic irrationals of the form
where , , and . To see this, let be a primitive -ideal, and set , which is a quadratic irrational, since by Theorem 2.3. By setting and , then and . Thus, to each primitive -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 and . Then is a primitive -ideal by Theorem 2.3, so to each quadratic irrational of type (2.11), there corresponds a primitive -ideal.

*Example 2.5. *It is possible to have a quadratic irrational of type (2.11) corresponding to a *non*primitive -deal. However, this does not alter the fact that there is a one-to-one correspondence between them and the primitive -deals, as demonstrated in Remark 2.4. For instance, the principal ideal is *not* primitive in since . Yet the quadratic irrational is of type (2.11). But corresponds to the primitive ideal 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 and we conisder the nonmaximal order , then the Diophantine equation with is solvable if and only if is a principal ideal in —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 , which is a quadratic irrational by Definition 2.1, but is not of type (2.11). Moreover, it corresponds to the ideal , which is not primitive, and it does not correspond to any primitive ideal as does above.

Remark 2.4 and Example 2.5 motivate the following.

*Definition 2.6 (ideals and quadratic irrationals). *To each quadratic irrational , with odd, even, (and ), there corresponds the primitive -ideal
We denote this ideal by and write for .

The infinite simple continued fraction of given by is called * periodic* (sometimes called * eventually periodic*), if there exists an integer and such that for all integers . We use the notation
as a convenient abbreviation. The smallest such natural number is called the * period length of *, and is called the *preperiod* of . If is the * least* nonnegative integer such that for all , then is called the * fundamental period* of . In particular, we consider the so-called *principal surd of *, for which it is known that

We will need the following facts concerning period length.

If is even, then and if is odd, then Furthermore, since we are assuming , then for , 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 be a quadratic irrational. If and , then is called *reduced*.

The next result sets the stage for our primary discussion.

Theorem 2.8 (pure periodicity equals reduction). *Let be an infinite simple continued fraction, with . Then is reduced if and only if is purely periodic, which means in (2.13), namely,
*

*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 -ideal is said to be *reduced* if it is primitive and does not contain any nonzero element such that both and .

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

Theorem 2.10 (reduced ideals and quadratic irrationals). * is reduced if and only if there is a such that with and .*

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

Corollary 2.11. *If is a primitive -ideal, with and , then is reduced.*

Now, we let be the ideal-class group of and the ideal class number. If are -ideals, then equivalence of classes in is denoted by , and the class of is denoted by . 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 , not just .)

Theorem 2.12 (the continued fraction algorithm). *Let be an -ideal corresponding to the quadratic irrational , and let be as given in (2.5)–(2.7). If , then for all . Moreover, there exists a least value such that is reduced for all .*

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

Corollary 2.13. *A reduced -ideal, for is principal if and only if for some positive integer in the simple continued fraction expansion of .*

*Proof. *See [13].

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

By Remark 2.14, once we have achieved a reduced ideal 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 is a reduced -ideal and is the least positive integer such that then for all have the same period length via
We denote this common value by where is the equivalence class of in , and call this value the* period length of the cycle of reduced ideals equivalent to *. If we wish to keep track of the specific ideal, then we write for .

*Remark 2.16. *If is a reduced -ideal, then the set
represents the* norms of all the reduced ideals equivelnt to * (via the continued fraction expansion of .

#### 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 is square-free with some integer , then .*

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

Corollary 3.2. *The only values for which with square-free are given by .*

In what follows, for , and is a square-free divisor of , with

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 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 is prime for all , then for some prime and . Also, the only values for which the above holds are
*

*Proof. *First we show that cannot be a perfect square. If , then
contradicting the hypothesis since .

Now we prove that must be square-free. If , then since is not a perfect square. Hence, . Also, if , then

a contradiction, so . Therefore, since
this contradicts the hypothesis if . Hence, and is square-free and so may be used for simple continued fraction expansions in the maximal order .

If is even, then

is composite unless , namely, unless , observing that since and . Thus, we may assume that is odd.

In the continued fraction expansion of , for all natural numbers by (2.6). We now show that .

Suppose that . By (2.18), for each , we may set

Since by (2.6), then . If for any , then by (2.17), , a contradiction. Thus, for . However, if , then is prime by hypothesis since for . We have, by (2.6), that
Therefore, where is prime if . Now suppose that . Since is even for all by (2.18), then and , a contradiction as above. We have shown that . If , then by the same argument , a contradiction. We have shown that . Hence, by (2.5),
which implies that forcing , a contradiction. We have shown that .

If , then , and with

contradicting the hypothesis unless which means . However, , where , contradicting the hypothesis. Hence, .

If , then by (3.8), for a prime , , and by (2.15). Hence, by (2.5)

which implies that . Now we show that .

Assume to the contrary that , since is odd by (2.18). Then by (2.6), . Therefore . If we let , then

by hypothesis, which forces , namely, , and . Therefore, , so , which implies that . Therefore,
contradicting the hypothesis which says is prime for all . We have shown that . Thus, , so by (3.8), for , we get , which means . For , (3.8) tells us that where is prime since . By (2.16), since , so by (2.6)
as we sought to show. Now we show that for these values.

By Theorem 2.12, if is an ideal class in , then contains a reduced ideal . Using a similar argument to the above on as we did for , we achieve that is in a cycle of period length , namely . Now in the simple continued fraction expansion of , let and . Then, as in the case for , (where we use the same symbols without risk of confusion since we are done with ), , , and by (3.8) applied to s values of , we must have that for some prime . If , then by Corollary 2.13, . If , then since is even by (2.18), we must have either and , or and . In either case, by Corollary 2.13 again, . Hence, .

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 where is prime, is prime for if and only if .*

Corollary 3.5. *Suppose that is prime, where is prime. Then all odd primes are inert in if and only if .*

Corollary 3.6. *Suppose that is prime, where is prime. Then for all positive integers and primes satisfying if and only if .*

Corollary 3.7. *Suppose that is prime, where is prime and with Dedekind-zeta function . Then if and only if .*

*Example 3.8. * A nice illustration of Corollary 3.7 is for , with where
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 is square-free for some odd integer , then .*

*Proof. *See [16].

Corollary 3.10. *If for square-free, then .*

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 is square-free, then for .*

*Proof. *See [9].

Corollary 3.12. *If for square-free, then .*

Theorem 3.13. *If is or prime for all where , then for , either
**
or
**
Also, the only values for which (3.16) holds are
**
and the only values for which (3.17) holds are
*

*Proof. *If , then by hypothesis , where is prime. Thus, , from which we deduce that the only possibility is and , namely, , contradicting the hypothesis. Thus, is not a square. Moreover, by the same argument as in the proof of Theorem 3.3, is square-free. Hence, we may apply continued fraction theory as above.

If is even, then , contradicting the hypothesis unless , for which . Hence we may assume that is odd and since gives which satisfies the hypothesis, we assume that is odd.

Let , so in the continued fraction expansion of , for by (2.6). If is given by (3.7), then by (3.8), we see that since , then by hypothesis

is prime for . In particular, for a prime , and for a prime . However, since is even for all by (2.18), then and is the only possibility. Thus, , so and . By the same argument as in the proof of Theorem 3.3, .

By virtually the same argument as used in the proof of Theorem 3.3, we get . However, by Corollary 3.12, the values of are those in the list (3.18).

Lastly, we may assume that , namely, . Again, by the same argument as used in the proof of Theorem 3.3, we get that , a prime, and . 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 , for which there exist exactly 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 .

We list the 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 Rabinowitsch polynomials below, excluding the degenerate case of which is included in the values in [7].

Note, as well, that although the original Theorem 1.1 only considers the values of , and Theorem 3.13 considers , the value of the Rabinowitsch polynomials therein also has being or prime as well. The restriction in Theorem 3.13 for the range of 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 , , then the followings are equivalent. *(a)* is or prime for all .*(b)* and is one of the following forms:(i) for some ,(ii) for a prime ,(iii) for a prime .*(c)

*.*

*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 and . The value is of their type (iii) with, in their notation, , , , and , so the corresponding Rabinowitsch polynomial is
The value is of type (iii) with , , , and , with Rabinowitsch polynomial
Indeed, is prime or for all or three times the length. Lastly, is of type (iii) with , , , and , with
Again, here is or prime for triple the length, namely, for . These 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 for a prime dividing . Recall that for a composite can occur only if where 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 , where are primes with , then the following are equivalent.(a) is or prime for all .(b) of for some and . *

*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
The use of is much less demanding than the use of and the lone two values found in [7] attest to this. However, they missed two other values for 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 with primes and is prime for all , then
Moreover, the only values for which (3.25) holds are
*

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 and . We now have a complete list of the Rabinowisch polynomials with one possible exception on the wide Richaud-Degert types, where we exclude for reasons given above. If we included the latter then the corrected list in [7] grows from to values.

*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.