On Hyperbolic Integers Hichem Gargoubi, Sayed Kossentini

To cite this version:

Hichem Gargoubi, Sayed Kossentini. On Hyperbolic Integers. 2021. hal-03198641

HAL Id: hal-03198641 https://hal.archives-ouvertes.fr/hal-03198641 Preprint submitted on 15 Apr 2021

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On Hyperbolic Integers

Hichem Gargoubi* and Sayed Kossentini

Abstract The algebra D of hyperbolic numbers is endowed with its standard f-algebra D structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of Z the ring of integers of D. We prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its lattice-ordered structure. We show that many of basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of ﬂoor and ceiling functions can be extended to Zh. In particular, Some properties of the hyperbolic Gaussian integers are obtained. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals and introduce the notion of toroidal groups.

Key words: Hyperbolic numbers, Hyperbolic Gaussian integers, f-ring, `-group, Lattice, Regular toroid.

1 Introduction

Motivated by solving problems concerning integers, number theorists of the XIX century have been interested in the analogous of Z when the concept of arithmetic can be developed. This leads Dedekind [3] to introduce and study the ring of integers OK of a number ﬁeld K. Usual notions of the arithmetic of Z, such as unique factorization into prime elements and Euclidean division, can be generalized, with various degrees, to such a ring.

In fact, when one studies the ring of integers Z it becomes clear to take into account the relation between its divisibility and its order properties as the existence of a unique positive gcd and the existence of unique factorization into product of positive primes. In this paper, the usual order structure of Z is seen as an Archimedean f-ring. Indeed, the general notion of f-algebra is simultaneously a Riesz space (or vector lattice) and an associative real algebra that fulfills certain ”positivity” conditions. A typical example of f-algebras is the linear space of real valued continuous functions on a topological space. Obviously, the fundamental example of Archimedean f-algebras is the field of real numbers. *Universitéde Tunis, I.P.E.I.T., Department of Mathematics, 2 Rue Jawaher Lel Nehru, Monfleury, Tunis, 1008 Tunisia. E-mail address: [email protected] Universitéde Tunis El Manar, Facultédes Sciences de Tunis, Department of Mathematics, 2092, Tunis, Tunisia. E-mail address: [email protected]

1 One of the points that motivates our approach is that there is no analogous of positivity in the ring OK . Actually, we prove a more general result concerning ring extensions of Q (see Theorem 3.1). This leads us to seek, in the direction hyperbolic numbers, a class of lattice-ordered rings (`-rings) of algebraic integers where we are able to generalize many of the basic divisibility and order properties of Z.

The hyperbolic numbers D (also called duplex numbers) are an extension of real numbers deﬁned in the same way that complex numbers C but with an imaginary unit j satisfying j2 = 1 (instead of i2 = −1) and so, D is not a division algebra. However, it enjoys an important order structure which makes it into the unique (up to an isomorphism) two dimensional Archimedean f-algebra. Therefore, basic notions of real analysis as sign, absolute value, Archimedean and Dedekind completeness are extended to the hyperbolic numbers [7]. Note that complex numbers and hyperbolic numbers are the only real commutative Cliﬀord algebras: ∼ ∼ D = ClR(1, 0) and C = ClR(0, 1)

The notion of partial order on D stimulates many authors and leads to interesting applications in different areas of mathematics. Alpay et al. [1] investigated the D-normed bicomplex modulus. In probability theory it is shown in [2] that Kolmogrov’s axioms and Bays theorems holds in the context of D-valued probabilities. Kumar et al. [15] intro- duced the notion of D-valued measure on a sigma algebra. As an Application to fractal geometry, a concept of Cantor sets in hyperbolic numbers was developed by Balankin et al. [4] and Téllez-Sánchez et al. [25]. Recently, the authors of the present paper used in [8] lattice-theoretical results to go further in the development of the theory of bicomplex zeta function. Further applications can be found in [13, 14, 17, 18, 20].

The purpose of this paper is to investigate a `-ring structure of algebraic integers from an arithmetic point of view. Our main result (Theorem 3.2) is the existence of a unique, up to ring isomorphism, Archimedean f-ring of quadratic integers called the ring of hyperbolic integers denoted Zh.

The present paper is organized in the following way: In section 2, we recall some notions and terminology concerning `-group, f-ring and f-algebra and we present basic notions and properties of hyperbolic numbers that will be used throughout this article. Section 3 introduces the lattice-ordered ring Zh of hyperbolic algebraic integers, and various of its properties are established. We introduce the notions of hyperbolic ﬂoor and ceiling functions which generalize that of real numbers. Section 4 and 5 are devoted to the divisibility in Zh. Many of the basic concepts of the arithmetic of Z are extended to Zh: the Euclidean division, the existence of a unique positive gcd, the existence of a unique factorization into a product of positive primes. In the last section, we establish some properties of the hyperbolic Gaussian integers as a subring of Zh. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals and introduce the notion of toroidal groups with hyperbolic indices.

2 2 Preliminaries

In this section, we recall basic properties of hyperbolic numbers which we will use throughout this paper (see [7]). First, we need to recall some notions and terminology of `-groups, f-rings [5, 23] and f-algebras [26]. A partially ordered group (G, +) is a partially ordered set under a partial order ≤ such that for any a, b ∈ G, a ≤ b implies a+c ≤ b+c and c+a ≤ c+b for all c ∈ G. A partially ordered group is a lattice-ordered group (an `-group) if the partial order is a lattice order (i.e., the supremum a ∧ b and the inﬁmum a ∨ b exist in G for all a, b ∈ G). Every element a in an `-group G can be written as a = a+ − a− where a+ = a ∨ 0 and a− = −a ∨ 0. The absolute value of a is deﬁned as |a| = a ∨ (−a) = a+ + a−. An `-group G is said to be Archimedean if for each nonzero a in G the set {na : n ∈ Z} has no upper bound in G; equivalently, a, b in G+ and na ≤ b for all n ∈ N implies a = 0. An Archimedean `-group is abelian. A real vectorial space V is said to be a vector lattice or Riesz space if V as a group is an `-group satisfying the property: a, b ∈ V , a ≤ b and α ∈ R+ implies αa ≤ αb. A ring R is an `-ring if R is an `-group and ab ∈ R+ for every a, b ∈ R+. An `-ring is Archimedean if its underlying group is Archimedean. An f-ring is an `-ring such that for every a, b ∈ R+, a∧b = 0 implies ac∧b = ca∧b = 0 for all c ∈ R+. An associative real algebra is an f-algebra if it is an f-ring and its underlying group is a vector lattice.

Hyperbolic numbers are deﬁned as the set

n 2 o D := z = x + jy : x, y ∈ R, j ∈/ R; j = 1 . It is a commutative ring with the group of units deﬁned as n o D∗ = z ∈ D : kzkh 6= 0 , where kzkh := zz¯ denotes the hyperbolic square-norm of z = x + jy,(x, y ∈ R) andz ¯ is the conjugate of z given by exchanging y ←→ −y. The hyperbolic plane has an important basis {e1, e2} where 1 + j 1 − j e = , e = . 1 2 2 2 e1 and e2 are mutually complementary idempotent zero divisors, i.e.,

2 2 e1 = e1; e2 = e2; e1 + e2 = 1; e1e2 = 0. (2.1) In this basis, each hyperbolic number z can be written as

z = π1(z)e1 + π2(z)e2, (2.2) where the maps π1, π2 : D → R are a pair of surjective ring homomorphisms deﬁned by

π1(x + jy) = x + y and π2(x + jy) = x − y.

3 From the representation (2.2), called the spectral decomposition [22], algebraic operations correspond to coordinate-wise operations, the square norm of z is the product π1(z)π2(z) and its conjugation is given by exchanging π1(z) ↔ π2(z). Moreover, we can deﬁne a partial order ≤ on D that makes it into Archimedean f-algebra, where

z, w ∈ D; z ≤ w if and only if πk(z) ≤ πk(w), k = 1, 2. (2.3) From this ordering the lattice operation are

z ∨ w = max{π1(z), π2(z)}e1 + max{π1(z), π2(z)}e2,

z ∧ w = min{π1(z), π2(z)}e1 + min{π1(z), π2(z)}e2. Moreover, z ∨w and z ∧w can be expressed as a I(D)-combination of z and w, where I(D) means the set of all idempotent elements of D. More precisely, this property is formulated in the following result.

Proposition 2.1 (Proposition 3.1 in [7]). For any z, w ∈ D there exist unique u, v ∈ I(D) satisfying uv = 0 and u + v = 1 such that z ∨ w = uz + vw and z ∧ w = vz + uw.

The Riesz space D is Dedekind complete that is, every nonempty subset A that is bounded above (resp. below), has a supremum sup A (resp. inﬁmum inf A), and

sup A = sup π1(A)e1 + sup π2(A)e2, (2.4)

inf A = inf π1(A)e1 + inf π2(A)e2. (2.5) In the ring of hyperbolic numbers there is a multiplicative group S called group of signs and given by n o ∼ S = 1, −1, j, −j = Z/2Z × Z/2Z. (2.6)

Theorem 2.1 (Theorem 5.1 in [7]). Let z ∈ D, then there exists an element ε ∈ S such that εz ≥ 0.

If kzkh 6= 0 then ε is unique, called sign of z, denoted sgn(z) and given by z sgn(z) = . |z|

The f-algebra D under the norm

n + o kzkR := min α ∈ R : α ≥ |z| = |z| ∨ |z| for all z ∈ D, is a unital Banach lattice algebra, i.e. the norm k.kR satisﬁes the properties: (i) |z| ≤ |w| implies kzkR ≤ kwkR; (ii) kzwkR ≤ kzkRkwkR and k1kR = 1. It follows that the exponential of z can be deﬁned, for any hyperbolic number z, as

∞ X zn ez := = eπ1(z)e + eπ2(z)e . n! 1 2 n=0

4 + Thus, the hyperbolic exponential function exp : D −→ D∗ is a group isomorphism that preserves conjugation and lattice operations. The hyperbolic logarithm function is given by the inverse isomorphism exp−1. Finally, let us mention that we will use the following notation: for any z, w ∈ D we write

+ z ≺ w if and only if w − z ∈ D∗ .

Therefore, if z, w ∈ R then z < w in R if and only if z ≺ w in D.

3 Hyperbolic integers

3.1 Basic deﬁnitions and properties

Let R be a ring extension of Q with degree n, i.e., a unital commutaive ring in which its underling group is a Q-vectorial space with dimension n. Each α ∈ R satisﬁes an equation

n n−1 α + an−1α + ··· + a0, ai ∈ Q.

If in addition the ai are integers, α is said to be algebraic integer. The set OR of all algebraic integers of R is a ring [21, Chap 2 ] called the ring of integers of R. In particular, a finite field extension K of Q is usually refereed as a number field. The ring OK of its integers is Notherian but in general is not a unique factorization domain see [10, Chap 12]. In lattice-ordered rings framework we obtain the following result.

Theorem 3.1. Let R be a ring extension of Q with degree 6= 1. If R is an integral domain then its ring of integers OR cannot be made into Archimedean f-ring.

Proof. Let R be an integral ring extension of Q with degree d > 1. Suppose that OR can be made into Archimedean f-ring. It follows from the identity x+x− = 0 holds for every x ∈ OR that OR is totally ordered Archimedean group, since it is an integral domain. Therefore, from H¨oldertheorem [5, Theorem 2.6.3], OR is isomorphic to a subgroup of R. This yields a contradiction since dimQR 6= 1. This complete the proof. The aim of this paper is the characterization of all Archimedean f-rings of quadratic integers. In fact, we show that, up to ring isomorphism, there is only one, namely the ring of integers of the extension Q(j) = {α + jβ ; α, β ∈ Q}. Theorem 3.2 (Hyperbolic integers). The ring of integers of Q(j), deﬁned by

Zh := Ze1 + Ze2, is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers called the ring of hyperbolic integers.

Proof. It is easy to check that Zh = Ze1 +Ze2 is the ring of integers of Q(j) = Qe1 +Qe2, and it is an Archimedean f-ring under the partial order induced from D. Then, its positive + + + cone is Zh = Z e1 + Z e2. Conversely, let OR be the ring of integers of a quadratic ring extension of Q. Suppose that OR is an Archimeadean f-ring, then it contains an element b

5 + − such that u1 = b 6= 0 and u2 = b 6= 0. Otherwise, OR is a totally ordered Archimedean group which implies (by H¨oldertheorem [5, Theorem 2.6.3]) that OR is isomorphic to a subgroup of R, which is a contradiction. Since OR is an f-ring, u1 and u2 are two disjoint nonzero elements such that u1u2 = 0. Moreover, the set {u1, u2} is independently linear over Q. Therefore, (v1, v2) := (α1u1, α2u2) is a basis of R where α1, α2 ∈ Q and verify 1 = α1u1 + α2u2. This means that R = Qv1 + Qv2, where the basis (v1, v2) satisﬁes the properties 2 2 1 = v1 + v2, v1v2 = 0 =⇒ v1 = v1, v2 = v2. (3.1)

Thus, from (3.1), αv1 +βv2 ∈ OR if and only α, β ∈ OQ = Z, that is OR = Zv1 +Zv2 with + + positive cone Z v1 + Z v2. It follows from (2.1) and (3.1) that the map ϕ : OR −→ Zh defined by ϕ(nv1 + mv2) = ne1 + me2 is a ring isomorphism. D Remark 3.1. Using notations of [21, Chap 2 ] one can define the set Z as the integral closure of Z in D i.e. the ring of algebraic integers of D. From Proposition 2.1, It D is clear that Z is a sublattice of D, under the induced partial order (2.3), and then an D R R Archimedean f-ring. One can easy check that Z = e1Z ⊕ e2Z . Indeed, an hyperbolic number α is an algebraic integer if and only if α1 = π1(α) and α2 = π2(α) are real algebraic integers. From this point of view the ring of hyperbolic integers Zh can be seen as D the smallest (with respect to inclusion) sub f-ring of Z containing {e1, e2}. √ As for the integers of a quadratic field Q( d), every hyperbolic integer a is the root of a monic polynomial P ∈ Z[X] given by 2 P (X) = X − 2Re(a)X + kakh, where kakh = aa¯ and Re(a) is the real part of a. However, Zh has zero divisors which are the multiples ne of e ∈ {e1, e2} with n ∈ Z \{0}. The units of Zh coincides with the group of signs of D (2.6), that is n o U(Zh) = S = 1, −1, j, −j .

2 In the hyperbolic plane D ≡ R , Zh is a ’square’ full lattice [19] with the fundamental parallelepiped P = {z ∈ D : 0 ≤ z ≺ 1} and minimal elements ±e1, ±e2.

Proposition 3.1. Let A be a nonempty subset of Zh. Then, the following holds. (i) If A is bounded from above and closed under ∨, then it has a largest element. (ii) If A is bounded from below and closed under ∧, then it has a smallest element. W Proof. (i) If A is ﬁnite, it is clear that max A = a∈A a. Otherwise, A is denumerable set which means that A can be viewed as a sequence (an). Let zn = a0 ∨ · · · ∨ an for n = 0, 1, ··· . Then, (zn)n≥0 is an increasing sequence of A which is bounded above. It follows that (zn) become constant, i.e., there exists an integer N ∈ N such that zn = zN for all n ≥ N. Hence max A = zN . (ii) We apply (i) for −A and using the duality formula inf A = − sup(−A).

6 Figure 1: Hyperbolic integers with fundamental parallelepiped P .

3.2 Ideals of Zh

In this subsection, we establish some properties involving ideals of Zh.

Proposition 3.2. For every ideal I in the ring Zh there exists a unique positive element gI such that I = gI Zh. Moreover, I is a sublattice of Zh.

Proof. Let I be an ideal of the ring Zh. Since for every k = 1, 2 the map πk is a surjective ring homomorphism from Zh to Z, then πk(I) is an ideal of the principal ideal domain Z. Therefore, there is a unique positive integer nk such that πk(I) = nkZ. Thus, the element gI = n1e1 + n2e2 generates I and it is the only positive one. It follows from Proposition 2.1 that I is a sublattice of Zh. Recall that an `-subgroup C (i.e., subgroup and sublattice) of an `-group G is said to be convex if 0 ≤ a ≤ b in G and b ∈ C imply a ∈ C. An `-ideal of an `-ring R is a convex `-subgroup of R that is also an ideal of R. The following characterizes `-ideals of the `-ring Zh.

Proposition 3.3. An ideal of Zh is an `-ideal if and only if it is generated by an idempotent element.

Proof. Let I be an ideal of Zh with positive generator gI . Proposition 3.2 means that I is an `-subgroup of Zh. So, I is an `-ideal if and only if I is convex. Suppose that gI is an idempotent element, i.e., gI ∈ {0, 1, e1, e2}. It is obvious that I is convex if gI = 0 or gI = 1. Assume that gI = e ∈ {e1, e2} that means I = eZh = eZ. Let a, b ∈ Zh be such that 0 ≤ a ≤ b and b ∈ I. Then a = αe for some real α, since eR is an order ideal of D

7 (see Theorem 3.5 in [7]). We must also have α ∈ Z, because a ∈ Zh. Hence, a ∈ I and this proves that I is convex. Conversely, assume that I is convex. We have to prove that the generator gI of I is an idempotent element. We distinguish two cases:

(i) if kgI kh = 0 the case gI = 0 is trivial. Suppose that gI = ne for some e ∈ {e1, e2} and some integer n ≥ 1. Then, we have 0 ≤ e ≤ ne with ne ∈ I which implies that e ∈ I = neZ. Thus, nk = 1 for some k ∈ Z. This yields that n = 1, and hence gI = e.

(ii) if kgI kh 6= 0, then 0 ≤ 1 ≤ gI with gI ∈ I which means that 1 ∈ I, i.e., I = Zh. Therefore, gI = 1.

Proposition 3.4. Let I be an ideal of Zh with the positive generator gI = n1e1 + n2e2. Then, Zh/I ' Z/n1Z × Z/n2Z.

In particular, Zh/I ' Z if and only if I is a nontrivial `-ideal.

Proof. Let I be an ideal in Zh with the positive generator gI = n1e1 + n2e2. One can easily see that the mapping Zh/I 3 a˙ 7→ π[1(a), π]2(a) ∈ Z/n1Z × Z/n2Z, establishes an isomorphism of Zh/I with Z/n1Z×Z/n2Z. In particular, from Proposition 3.3, Zh/I ' Z if and only if I is a nontrivial `-ideal.

3.3 Hyperbolic ﬂoor and ceiling functions

+ − Let z ∈ D. Then the sets E (z) := {k ∈ Zh : k ≤ z} and E (z) := {k ∈ Zh : k ≥ z} are two nonempty sublattices of Zh. Thus from Proposition 3.1, the notions of ﬂoor b.c and ceiling d.e functions on real numbers can be extended to the hyperbolic numbers in the following way.

Deﬁnition 3.1. The functions b.cD and d.eD from D to Zh deﬁned by

n o bzcD := max k ∈ Zh : k ≤ z , n o dzeD := min k ∈ Zh : k ≥ z are called, respectively, hyperbolic ﬂoor function and hyperbolic ceiling function.

By (2.4) and (2.5), we derive that

bzcD = bπ1(z)ce1 + bπ2(z)ce2 and dzeD = dπ1(z)ee1 + dπ2(z)ee2. (3.2)

8 Therefore

z − 1 ≺ bzcD ≤ z ≤ dzeD ≺ z + 1 for all z ∈ D. (3.3) Let as consider a hyperbolic closed interval (see [4] and [25] ) deﬁned by

[α, β] := {z ∈ : α ≤ z ≤ β}. D D Geometrically, [α, β] is a rectangle (Figure 2) when kα − βk 6= 0 called nondegenerate D h interval, otherwise it is a line segment [α, β] parallel to one of the two bisector axis . The open interval (α, β) and half-open intervals (α, β] and [α, β) are deﬁned in a D D D

Figure 2: Non degenerate hyperbolic closed interval [α, β] . D similar way replacing ≤ by ≺ in left-right and left/right, respectively. However, all these intervals are empty if kα − βk = 0. One has (α, β) = [α, β] \ (∂ ∪ ∂ ), (α, β] = h D D α β D [α, β] \ ∂ and [α, β) = [α, β] \ ∂ where ∂ and ∂ are the two edges that meet, D α D D β α β respectively, at α and β.

As on real numbers, the functions b.cD and d.eD allows one to determine the number

NZh (I) of hyperbolic integers in a hyperbolic interval I by considering the four types above.

Proposition 3.5. Let α, β ∈ D be such that α ≤ β then

NZh ([α, β] ) = bβcD − dαeD + 1 , D h

NZh ([α, β) ) = dβeD − dαeD , D h

NZh ((α, β] ) = bβcD − bαcD , D h

NZh ((α, β) ) = dβeD − bαcD − 1 . D h Proof. Let us denote by I = [α, β] , I = [α, β) , I = (α, β] and I = (α, β) . Then 1 D 2 D 3 D 4 D 2 2 the sets Zh ∩ Ik are bijectively mapped onto Z ∩ ϕ(Ik) via the map from D to R deﬁned

9 2 by ϕ(z) = (π1(z), π2(z)). Thus, for k = 1,..., 4, NZh (Ik) = #Z ∩ Rk where Rk = ϕ(Ik) are the rectangles

R1 = [π1(α), π1(β)] × [π2(α), π2(β)] ,R2 = [π1(α), π1(β)) × [π2(α), π2(β))

R3 = (π1(α), π1(β)] × (π2(α), π2(β)] ,R4 = (π1(α), π1(β)) × (π2(α), π2(β)) .

Therefore,

NZh (I1) = (bπ1(β)c − dπ1(α)e + 1) (bπ2(β)c − dπ2(α)e + 1) ,

NZh (I2) = (dπ1(β)e − dπ1(α)e)(dπ2(β)e − dπ2(α)e) ,

NZh (I3) = (bπ1(β)c − bπ1(α)c)(bπ2(β)c − bπ2(α)c) ,

NZh (I4) = (dπ1(β)e − bπ1(α)c − 1) (dπ2(β)e − bπ2(α)c − 1) .

Finally, the results follows from (3.2) and the propriety kzkh = π1(z)π2(z).

4 Divisibility

4.1 First properties

Divisibility in Zh is deﬁned in a natural way: we say b divides a , or a is a multiple of b (and write b|a) if a = bc for some c ∈ Zh. In this case, we call b a divisor of a.

Proposition 4.1. For a, b ∈ Zh we have

(i) a|b in Zh implies kakh kbkh ∈ Z;

(ii) a|b and kbkh 6= 0 implies |a| ≤ |b|; (iii) a|b and b|a if and only if |a| = |b|. Proof. (i) If a|b then b = ac which implies, by multiplicity of k.k , that kbk = kak kck . h h h h

Hence, kakh kbkh ∈ Z. (ii) If a|b and kbkh 6= 0, then b = ca for some c ∈ Zh with kckh 6= 0 that means |c| ≥ 1. Therefore, |b| − |a| = (|c| − 1)|a| ≥ 0. (iii) a|b and b|a if and only if a = εb for some unit ε, i.e., (by Proposition 5.1 in [7]) if and only if |a| = |b|.

4.2 Hyperbolic euclidean division and congruence

Theorem 4.1. Let a, b ∈ Zh with kbkh 6= 0, then there exist unique q, r ∈ Zh such that

a = bq + r, 0 ≤ r ≺ |b|.

The hyperbolic integers q and r are called, respectively, the quotient and the remainder of the division of a by b.

10 Proof. We consider ﬁrst the uniqueness. Assume that

a = bq1 + r1 = bq2 + r2, 0 ≤ r1, r2 ≺ |b|.

Then, |r − r | 0 ≤ |q − q | = 1 2 ≺ 1. 1 2 |b|

Hence, q1 = q2 which implies r1 = r2. Consider now the existence. Put $ % a q = ε and r = a − bq, |b| D |b| where ε = sgn(b) = b and b.cD is the hyperbolic ﬂoor function (3.1). Then, we have q, r ∈ Zh and a = bq + r. It remains to prove that 0 ≤ r ≺ |b|. From (3.3) one has $ % a a a − 1 ≺ ≤ . (4.1) |b| |b| |b| D Multiply (4.1) by −εb = −|b| and use r = a − bq to get the desired inequality.

As for integers, congruences in Zh are deﬁned using divisibility.

Deﬁnition 4.1. Let a, b, v ∈ Zh, we write a ≡ b mod v if and only if v|(a − b).

Since congruence modulo 0 means equality and a|b if and only if |a| b, we usually assume the modulus is a nonzero positive element of Zh.

Proposition 4.2. For a, b, c, v ∈ Zh one has (i) a ≡ b mod v and c ≡ d mod v implies a + c ≡ b + d mod v and ac ≡ bd mod v;

(ii) if also a, b, v ∈ Z then a ≡ b mod v in Zh if and only if a ≡ b mod v in Z;

(iii) a ≡ b mod v in Zh if and only if πk(a) ≡ πk(b) mod πk(v) in Z for k = 1, 2; (iv) a ≡ b mod v if and only if a¯ ≡ ¯b modv. ¯

Proof. Straightforward.

Let v ∈ Zh, v 0. Then (by proposition 3.4), the number of the residue classes modulo v is its square norm kvkh. For instance, the four binary classes are the set ˆ ˆ Zh/2Zh = {0, 1, eb1, eb2}. (4.2)

11 4.3 Positive gcd and positive lcm

According to Proposition 3.2, for every a, b ∈ Zh the ideals aZh + bZh and aZh ∩ bZh are generated by a unique positive element. This justiﬁes the following result.

Theorem 4.2. Every a, b ∈ Z have a unique positive greatest common divisor gcd (a, b) h Zh and a unique positive latest common multiple lcmZh (a, b). Moreover,

gcd (a, b) = gcd(π (a), π (b))e + gcd(π (a), π (b))e , Zh 1 1 1 2 2 2

lcmZh (a, b) = lcm(π1(a), π1(b))e1 + lcm(π2(a), π2(b))e2 .

As an immediate consequence of Theorem 4.2 we have the following properties of the gcd and lcm which are extension of the corresponding ones in . Zh Zh Z

Proposition 4.3. For a, b ∈ Zh we have (i) gcd (|a|, |b|) = gcd (a, b) and lcm (|a|, |b|) = lcm (a, b); Zh Zh Zh Zh (ii) gcd (a, b) lcm (a, b) = |ab|; Zh Zh (iii) gcd (a, b) = gcd (¯a, ¯b) and lcm (a, b) = lcm (¯a, ¯b); Zh Zh Zh Zh (iv) gcd (a ∨ b, a ∧ b) = gcd (a, b) and lcm (a ∨ b, a ∧ b) = lcm (a, b), Zh Zh Zh Zh + Remark 4.1. In view of Proposition 4.1, the quazi-order on Zh deﬁned by a ≤ b if and only if a|b,

+ is a partial order, and under such order, Zh is a lattice ordered multiplicative monoid with a ∧ b = gcd (a, b) and a ∨ b = lcm (a, b) for all a, b ∈ Z+. Zh Zh h

5 Primes and irreducibles in Zh

In this section we characterize all prime and irreducible elements of Zh. We also extend the unique factorization theorem of Z to Zh. The set of all prime numbers: 2, 3, 5, 7, 11, ··· will be denoted by P. For basic notions and terminology about prime and irreducible elements we refer to [16, Chap II].

5.1 Characterization Theorem 5.1 (Hyperbolic primes). The following statements are satisﬁed.

e1 e2 (i) Prime elements of Zh are u = εv where ε is a unit and v ∈ {e1, e2, p , p : p ∈ P}.

(ii) Irreducible elements of Zh are u = εv where ε is a unit and v is in

e1 e2 Ph := {p , p : p ∈ P}.

where Ph is deﬁned as the set of hyperbolic prime numbers (or hyperbolic primes) ,

12 Proof. Let v = ne1 + me2 be a nonzero and nonunit positive element of Zh. (i) By Proposition 3.4, v is prime if and only if Z/n Z × Z/m Z is an integral domain, i.e, if and only if one ring is zero and the other is an integral domain. It follows that (n, m) ∈ e1 e2 {(1, 0), (0, 1), (p, 1), (1, p): p ∈ P}. Which means that v ∈ {e1, e2, p , p : p ∈ P}. (ii) If v is irreducible, then vZh is a maximal ideal, since by Proposition 3.2, every ideal in the ring Zh is principal. Therefore, vZh is a prime ideal which means that v is prime. e So from (i), either v = e1 or v = e2 or v = p for some prime number p and some e e ∈ {e1, e2}. We will prove that p is irreducible, since the atoms e1 and e2 are not. Let e a, b ∈ Zh be such that p = ab. Then, taking the norm k.kh, we obtain p = kakhkbkh. Since p is irreducible in Z it follows that kakh = ±1 or kbkh = ±1. Hence, either a ∈ S or b ∈ S.

Remark 5.1. Theorem (5.1) shows that hyperbolic primes are positive non-zero-divisor prime elements of Zh.

5.2 Unique factorization theorem The fundamental theorem of arithmetic states that every nonzero integers n can be written uniquely in the form Y n = ε pvn(p), p∈P where ε is a unit ( ε = sgn(n)) and vn : P −→ N with vn(p) 6= 0 for a ﬁnite number of p. The following statements shows that this property can be generalized to hyperbolic integers.

Theorem 5.2. Every a ∈ Zh with kakh 6= 0 can be written uniquely in the form Y a = ε pva(p), p∈P

+ where ε is a unit and va : P −→ Zh with va(p) 6= 0 for a ﬁnite number of p.

Proof. Let a ∈ Zh with kakh 6= 0. By Theorem 2.1, there is a unique unit ε ∈ S such that a = ε|a|. Let n1, n2 ∈ N be such that |a| = n1e1 +n2e2. Then, n1, n2 6= 0. By fundamental theorem of arithmetic, for every n ∈ N there is a unique application µn : P −→ N with µa(n) = 0 for almost all p such that Y n = pµn(p). p∈P

+ Let va : P −→ Zh be the function deﬁned by va(p) = µn1 (p)e1 + µn2 (p)e2. Therefore Y |a| = pva(p). p∈P

13 Theorem 5.3 (Unique factorization theorem). Every a ∈ Zh with kakh 6= 0 can be written uniquely in the form Y a = ε uva(u),

u∈Ph where ε is a unit and va : Ph −→ N with va(u) = 0 for almost all u. Proof. The proof follows immediately from Theorem 5.2 by observing that hyperbolic primes are pe1 and pe2 with p ∈ P ( Theorem 5.1).

6 Hyperbolic Gaussian integers

By analogy to complex numbers, the hyperbolic Gaussian integers or more simply the h-Gaussian integers (also called split Gaussian integers [11]) are the set n o Gh = Z[j] := x + jy : x, y ∈ Z .

It is a subring of Zh with zero divisors that are the multiples n(1 ± j), n ∈ Z \{0}, and units that are 1, −1, j, and −j. From the four binary classes (4.2) of hyperbolic integers we have the following characterization of h-Gaussian integers.

Theorem 6.1. Let a ∈ Zh, then a ∈ Gh if and only if either a ≡ 0 mod 2 or a ≡ 1 mod 2. n+m n−m Proof. Let a = ne1 + me2 = 2 + j 2 ∈ Zh with n, m ∈ Z. So, a ∈ Gh if and only if n ≡ m mod 2, i.e., if and only if either a ≡ 0 mod 2 or a ≡ 1 mod 2.

In view of Theorem 2.1 and by the fact that the units of Gh are the set S one can see that Gh is closed under absolute value. But, it is not an `-subgroup of Zh, since 0 ∨ j = e1 ∈/ Gh. However, the next result gives under which condition the supremum of two incomparable (with respect to the order induced by Zh) h-Gaussian integers exists in Gh.

Proposition 6.1. Let a, b ∈ Gh be incomparable, then

a ∨ b ∈ Gh if and only if a ≡ b mod 2.

Proof. Let a, b ∈ Gh be incomparable. Then, from Proposition 2.1 we can write a ∨ b = ua + vb, for some u, v ∈ {e1, e2} with u + v = 1. Therefore, Theorem 6.1 implies that a ∨ b ∈ Gh if and only if a ≡ b mod 2.

Proposition 6.2. Every a ∈ Gh with kakh 6= 0 can be written uniquely in the form

ν Y va(p) a = εa2 p , p6=2

+ + where εa is a unit, va : P −→ Zh with va(p) = 0 for almost all p and ν ∈ Zh is such that ν = 0 if a ≡ 1 mod 2 and ν 0 if a ≡ 0 mod 2.

14 Proof. From Theorem 5.2, a can be uniquely expressed in the form

Y va(p) ν Y va(p) a = εa p = εa2 p , p p6=2

va + + where εa is a unit and P −→Zh with va(p) = 0 for almost all p and ν = va(2) ∈ Zh . Therefore, a ≡ 2ν mod 2 since, pva(p) ≡ 1 mod 2 for p 6= 2 . It follows that ν = 0 if a ≡ 1 mod 2 and ν 0 if a ≡ 0 mod 2.

7 Applications

7.1 Dirichlet divisor problem P The Dirichlet divisor problem, arises from estimating D(ρ) := n≤ρ d(n), where d(n) is the number of positive divisors of n. A well known result is D(ρ) = ρ ln ρ+(2γ−1)ρ+∆(ρ), where γ is Euler’s constant and ∆(ρ) is the error term. The Dirichlet divisor problem asks for the correct order of magnitude of ∆(ρ) as ρ −→ ∞ (see e.g. [12, Chap 5]). From a geometrical point of view D(ρ) is equal to the number of lattice points in the ﬁrst quadrant under the hyperbola xy = ρ. Thus, this is equivalent to determine the number of hyperbolic integers a 0 with kakh ≤ ρ, i.e.,

+ D(ρ) = #Zh ∩ D (ρ), D(ρ) := {z ∈ D∗ : kzkh ≤ ρ}. (7.1) Deﬁne D (ρ) = D(ρ) ∩ [−ρ, ρ] . ? D Geometrically, D (ρ) is the square [−ρ, ρ] if ρ ≤ 1, and D (ρ) [−ρ, ρ] if ρ > 1 as ? D ? ( D represented in Figure 3.

√ 1 j ln ρ Figure 3: A representation of D?(ρ) for ρ > 1 with τ = ρe 2 .

15 + Proposition 7.1. We have D(ρ) = #Zh ∩ D? (ρ). + + Proof. It suﬃces, from (7.1), to prove that Zh ∩ D (ρ) = Zh ∩ D? (ρ). Then, suppose that Z ∩ ([−ρ, ρ]c ∩ D+(ρ)) 6= ∅. (7.2) h D Observing that [−ρ, ρ] is the closed ball B (0, ρ) in ( , k.k ) where k.k is the lattice D R D R R norm (2), then Eq (7.2) yields that there exists a hyperbolic integer a = ne1 + me2 0 such that kakR = max{n, m} > ρ and kakh = nm ≤ ρ. Therefore, from the identity nm = max{n, m} min{n, m} one has that ρ ≥ nm > ρ. Which is a contraction. Hence,

+ + Zh ∩ D (ρ) = Zh ∩ D? (ρ).

Let n be an integer ≥ 2. Deﬁne ξk, λk, µk and ηk such that

√ jk ln ρ √ ξk = ρe 2n , λk = ξk ∧ ρ, for k = 0, ··· , n;

ηk = ξk ∨ ξk−1, µk = ξk ∧ ξk−1, for k = 1, ··· , n. Thus, and referring to Figure 4, proposition 7.1 yields that

Figure 4: .

− + Dn (ρ) ≤ D(ρ) ≤ Dn (ρ), (7.3) where k=n √ X D−(ρ) = N (0, ρ] + 2N ((α, τ] ) + 2 N ((λ , ξ ] ) , (7.4) n Zh D Zh D Zh k k−1 D k=2 k=n √ X D+(ρ) = N (0, ρ] + 2N ((α, τ] ) + 2 N ((λ , η ] ) . (7.5) n Zh D Zh D Zh k k D k=1

16 − + Let ∆n (ρ) and ∆n (ρ) be such that − − ∆n (ρ) = Dn (ρ) − (ρ ln ρ + (2γ − 1)ρ), + + ∆n (ρ) = Dn (ρ) − (ρ ln ρ + (2γ − 1)ρ). √ Let δ(ρ) be a function deﬁned for ρ > 1 by δ(ρ) = 0 if ρ∈ / N and δ(ρ) = d(ρ) + χN( ρ), otherwise. + − Proposition 7.2. One has lim sup(∆n (ρ) − ∆n (ρ)) ≤ δ(ρ). n→∞ Proof. We have

k=n X ∆+(ρ) − ∆−(ρ) = D+(ρ) − D−(ρ) = 2 N ((µ , η ] ) . n n n n Zh k k D k=1 ∗ ∗ √ jt 1 Denote by J = Zh∩(D?(ρ)\γρ ) where γρ is the image of γρ(t) = ρe deﬁned in 0, 2 ln ρ . ∗ Since J is a nonempty ﬁnite set one obtains dγ∗ = min d(h, γ ) = min inf kh − ξk > 0. ρ ρ ∗ h∈J h∈J ξ∈γρ Let N be an integer such that √ ln ρ p 2 ρ sinh( ) cosh(ln ρ) < d ∗ . 4N γρ Then, for every n ≥ N and for every k = 1, ··· , n we have

diam [µk, ηk] = kξk − ξk−1k √ ln ρ ≤ 2 ρ sinh( )pcosh(ln ρ) 4n √ ln ρ ≤ 2 ρ sinh( )pcosh(ln ρ) 4N ∗ < dγρ

Thus, if Zh ∩ (µk, ηk] 6= ∅ then for ever h ∈ Zh ∩ (µk, ηk] we have ∗ ∗ d(h, γk) ≤ diam [µk, ηk] < dγρ h i where γ∗ = γ (k−1) ln ρ, k ln ρ . Therefore, Z ∩ (µ , η ] ⊂ Z ∩ γ∗. Hence k ρ 2 2 h k k D h ρ n [ Z ∩ (µ , η ] ⊂ Z ∩ γ∗. h k k D h ρ k=1 It follows from the inclusion above that + − ∗ ∆n (ρ) − ∆n (ρ) ≤ 2#Zh ∩ γρ for all n ≥ N. Therefore + − + − ∗ lim sup(∆n (ρ) − ∆n (ρ)) ≤ sup(∆n (ρ) − ∆n (ρ)) ≤ 2#Zh ∩ γρ . n→∞ n≥N ∗ + Thus, it follows from Zh ∩ γρ = {h ∈ Zh , Im(h) ≥ 0 : khkh = ρ} that ∗ 2 2#Zh ∩ γρ = 2#{(x, y) ∈ Z , 0 ≤ x ≤ y : xy = ρ} = δ(ρ). This complete the proof.

17 Proposition 7.3. We have 0 ≤ ∆(ρ) − ∆−(ρ) ≤ δ(ρ) where k=n ! − X 1 + (k−1) 1 1 − (k−1) 1 − k ∆ (ρ) = ψ(ρ) + 2 lim inf bρ 2 2n c − bρ 2 c bρ 2 2n c − bρ 2 2n c , n→∞ k=2 1 2 1 ψ(ρ) = bρ 2 c + 2 bρc − bρ 2 c − (ρ ln ρ + (2γ − 1)ρ). Proof. We have − + − 0 ≤ ∆(ρ) − ∆n (ρ) ≤ Dn (ρ) − Dn (ρ). − − Put ∆ (ρ) = lim inf ∆n (ρ). Thus, from proposition 7.2 one has n→∞ − − 0 ≤ lim sup(∆(ρ) − ∆n (ρ)) = ∆(ρ) − ∆ (ρ) ≤ δ(ρ). n→∞ From the above inequality we have k=n X ∆−(ρ) = ψ(ρ) + 2 N ((λ , ξ ] ) , n Zh k k−1 D k=2 √ ψ(ρ) = N (0, ρ] + 2N ((α, τ] ) − (ρ ln ρ − (2γ − 1)ρ). Zh D Zh D Straightforward calculations give 1 τ = ρe1 + e2, α = ρ 2 e1 , 1 n−k n+k n−k n+k n−(k−1) λk = ρ 2 e1 + ρ 2n e2, ξk = ρ 2n e1 + ρ 2n e2, ηk = ρ 2n e1 + ρ 2n e2 . Therefore, from Proposition 3.5 we have √ 1 2 N (0, ρ] = bρ 2 c , Zh D 1 N ((α, τ] ) = bρc − bρ 2 c , Zh D n+(k−1) 1 n−(k−1) n−k N ((λ , ξ ] ) = bρ 2n c − bρ 2 c bρ 2n c − bρ 2n c . Zh k k−1 D Hence k=n X 1 (k−1) 1 1 (k−1) 1 k − 2 + 2n 2 2 − 2n 2 − 2n ∆n (ρ) = ψ(ρ) + 2 bρ c − bρ c bρ c − bρ c , k=2 1 2 1 ψ(ρ) = bρ 2 c + 2 bρc − bρ 2 c − (ρ ln ρ − (2γ − 1)ρ). Finally k=n ! − X 1 + (k−1) 1 1 − (k−1) 1 − k ∆ (ρ) = ψ(ρ) + 2 lim inf bρ 2 2n c − bρ 2 c bρ 2 2n c − bρ 2 2n c . n→∞ k=2

Theorem 7.1. For every real > 0, we have k=n ! X 1 + (k−1) 1 1 − (k−1) 1 − k ∆(ρ) = ψ(ρ) + 2 lim inf bρ 2 2n c − bρ 2 c bρ 2 2n c − bρ 2 2n c + O(ρ ). n→∞ k=2 Proof. From proposition 7.3 ∆(ρ) is given by ∆−(ρ), and the error is the order of δ(ρ). Thus, the proof follows from the deﬁnition of δ by observing that d(n) = O(n) for every > 0.

18 7.2 Toroidal groups An ordinary polyhedron topologically torus-like T 2 = S1 × S1 is called a toroid. Then, its Euler number v − e + f = 0. A toroid is said to be regular if each face has exactly a edges, and at each vertex exactly b edges meet. Thus, from Euler’s formula we get three classes of regular toroids: A class T1 with a = 3, b = 6, a class T2 with a = b = 4 and a class T3 with a = 6 , b = 3. More about toroids can be found in [24]. Let us recall some results from [8]. Bicomplex numbers are the set n o B := x + yi + zj + tk : x, y, z, t ∈ R; i, j, k ∈/ R , where i, j, k are imaginary units that satisfy the following multiplication rules

i2 = k2 = −1, j2 = 1, ik = ki = j.

It is a normal complexiﬁed f-algebra B = D + iD, where√ the modulus of ω = a + ib ∈ B is given by |ω| := sup{a cos θ + b sin θ : 0 ≤ θ ≤ 2π} = a2 + b2 ∈ D+. The unit D-sphere 2 SD = {ω ∈ B : |ω| = 1} is a multiplicative group homeomorphic to the torus T via the 2iπz map f : D/Zh −→ SD :z ˆ 7→ e . The bicomplex exponential function Exp is a group homomorphism from B to B∗ with ker(Exp) = 2iπZh. Using idempotent hyperbolic numbers e1 and e2, the set B can be decomposed as

B = Ce1 ⊕ Ce2. From this representation the algebraic operations can be deﬁned by component-wise op- ω z1 z2 erations. Moreover, the exponential of ω = z1e1 + z2e2 ∈ B is given by e = e e2 + e e2. In this section we show that the set of vertices of a toroid is a multiplicative group isomorphic to the group of the bicomplex νth-roots of the unity, where ν is a hyperbolic integer.

Proposition 7.4. Let ν ≥ 1 be an hyperbolic integer. Then the νth-roots of unity are the set 2iπh U = {e ν : h ∈ [0, ν − 1] }. ν Zh

It is a multiplicative group with order kνkh.

Proof. Let ν = n1e2 + n2e2 ≥ 1 be an hyperbolic integer, and ω = z1e1 + z2e2 ∈ B. Then 2iπhk ν n th ω = 1 if and only if zk = e k , hk ∈ [0, nk − 1], (k = 1, 2). Therefore, the ν -roots of 1 are the set 2iπh U = {e ν : h ∈ [0, ν − 1] }. ν Zh It follows from hyperbolic division algorithm (Theorem (4.1)) and by the fact that ker(Exp) = 2iπZ that U is a multiplicative group. Moreover, |U | = # [0, ν − 1] = kνk . h ν ν Zh h Proposition (7.4) implies that the map

σ : [0, ν − 1] −→ U Zh ν 2iπh h 7−→ e ν

19 is a bijection.

Now denote VP the set of all vertices of a toroid P . Then, we have n1 parallels Γ0, Γ1, ··· , Γn1−1 and n2 meridians Ψ0, Ψ1, ··· , Ψn2−1 in which for every V ∈ VP there exist unique pair

(h1, h2) of integers with 0 ≤ h1 ≤ n1 − 1 and 0 ≤ h2 ≤ n2 − 1 such that V ∈ Γh1 ∩ Φh2 . Put h = h1e1 +h2e2, write V = Vh and ν = n1e1 +n2e2. This means that Vp is bijectively mapped onto [0, ν − 1] via the map Zh ξ : V −→ [0, ν − 1] p Zh Vh 7−→ h

The pair (n1, n2) will be called type of P .

Proposition 7.5. Let P be a toroid with type (n1, n2) and let ν = n1e1 + n2e2 and ϑ = σ ◦ ξ. Then the set VP endowed by the product deﬁned by

−1 Vh.Vh0 = Vl, l = σ (ϑ(Vh)ϑ(Vh0 )). is a group isomorphic to Uν.

Proof. It is clear that ϑ = σ ◦ ξ is a bijection from VP to Uν. It is easy to verify that V endowed by the above multiplication is a group with the identity V and the inverse P 0 V −1 = ϑ−1 1 . We will prove that ϑ is a group homomorphism. One has that h ϑ(Vh) ϑ(V0) = 1 and

ϑ (Vh.Vh0 ) = ϑ(Vl)

= σ (ξ(Vl)) = σ(l) −1 = σ σ (ϑ(Vh)ϑ(Vh0 ))

= ϑ(Vh)ϑ(Vh0 ).

Deﬁnition 7.1. Let P be a toroid with type (n1, n2) and let ν = n1e1 + n2e2. The set VP endowed with the product deﬁned above is called toroidal group with type (n1, n2) and denoted by T (n1, n2). The group Uν = ϑ(T (n1, n2)) is called the bicomplex representation of T (n1, n2).

Remark 7.1. (i) The groups T (n1, n2) and T (n2, n1) are isomorphic by conjugation, ¯ 2iπ h 2iπ h since the mapping e ν 7−→ e ν¯ is a group isomorphism from Uν to Uν¯.

th (ii) The case n1 = n2 was considered in [8] to describe The n -roots of unity, with n ∈ N. An example of two conjugate toroidal groups is illustrated by Figures 5 and 6.

20 Figure 5: A representation of the toroidal group T (5, 7) in class T2.

Figure 6: A representation of the toroidal group T (7, 5) in class T2.

References

[1] Alpay, D., Luna-Elizarrar´as,M.E., Shapiro,M., Struppa, D.C.: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis. Springer Science & Business Media (2014)

[2] Alpay, D., Luna-Elizarrar´as,M.E., Shapiro,M.: Klomogrov’s axioms for probabilities with values in hyperbolic numbers. Adv. Appl. Cliﬀord Algebra. 27(2), 913-929 (2017)

[3] Dedekind, R. Theory of algebraic integers. Cambridge University Press, 1996 ( A translation of Uber¨ die theorie der ganzen algebraischen zahlen, 1932)

[4] Balankin, A.S., Bory-Reyes, J., Luna-Elizarrar´as,M.E., Shapiro,M.: Cantor-type sets in hyperbolic numbers. Fractals 24 (4) (2016) 1650051

[5] Bigard, A., Keimel,K., Wolfenstein, S.: Groupes et Anneaux R´eticul´es.Lecture Notes in Mathematics, vol. 608. Springer, Berlin (1977)

[6] Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampatti, E.: The Mathematics of Minkowski Space-Time, Birkh¨auser,Verlag, Basel (2008)

[7] Gargoubi, H., Kossentini, S.: f-Algebra Srtucture On Hyperbolic Numbers. Adv. Appl. Cliﬀord Algebras 26(4),1211-1233 (2016)

[8] Gargoubi, H., Kossentini, S.: Bicomplex numbers as a normal complexiﬁed f-algebra, to appear in Communications in Mathematics.

21 [9] Guy, R.: Unsolved problems in number theory (Vol. 1). Springer Science & Business Media (2004)

[10] Ireland, K., Rosen, M. : A Classical Introduction to Modern Number Theory. Second Edition Springer Science & Business Media New York (1982)

[11] Cifuente, J. C., Strapasson, J. E., Corrˆea,A. C., & Kitani, P. M. (2007). The Ring of Integers in the Canonical Structures of the Planes. arXiv preprint arXiv:0707.0700.

[12] Kr¨atzel,E.: Lattice points (Vol. 33). Springer Science & Business Media (1989).

[13] Kumar, R., Singh, K., Saini, H., and Kumar, S.: Bicomplex Weighted Hardy Spaces and Bicomplex C∗-algebra. Adv. Appl. Cliﬀord Algebra. 26(1),217-235 (2016)

[14] Kumar, R. and Saini, H.: Topological Bicomplex modules. Adv. Appl. Cliﬀord algebra. 26(4),1249-1270 (2016)

[15] Kumar, R., Sharma, K., Tundup, R., Wazir, S.: Orlicz Spaces with Bicomplex Scalars. arXiv: 1401.7112v2 Jan (2017)

[16] Lang, S.: Algebra, Undergraduate Texts in Mathematics 211, Springer (2005)

[17] Luna-Elizarraras M.E. , C.O. Perez-Ragalado, C.O., Shapiro, M.: On linear function- als and Hahn-Banach theorems for hyperbolic and bicomplex modules, Adv. Appl. Cliﬀord Algebra. 24, 1105-1129 (2014)

[18] Luna-Elizarraras M.E. , C.O. Perez-Ragalado, C.O., Shapiro, M.: On the Laurent series for bicomplex holomorphic functions: Complex Var. Elliptic Equ. (2017) , doi: 10.1080/17476933.2016.1250404

[19] Martinet, J.: Perfect Lattices in Euclidean Spaces. Springer-Verlag, Berlin (2003)

[20] Rochon, D., Tremblay, S.: Bicomplex quantum mechanics II. The Hilbert Space. Adv. Appl. Cliﬀord Algebra. 16 (2), 135-157 (2006)

[21] Samuel, P.: Th´eorieAlg´ebriquedes nombres. Hermann & Cie, Paris(1967)

[22] Sobczyk, G.: The hyperbolic number plane. Coll. Math. J. 26 (4), 268-280 (1995)

[23] Steinberg, S.: Lattice-ordered Rings and Modules. Springer, Dordrecht (2010)

[24] Szilassi, L.: Regular Toroids. Structural topology (1986)

[25] T´ellez-S´anchez, G.Y., Bory-Reyes, J. : More about Contor like sets in hyperbolic numbers. Fractals 25 (5) (2017) 1750046

[26] Zaanen, A.C.: Riesz Spaces II. North-Holland. Amsterdam (1983)