EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6781 ISSN 1307-5543 – ejpam.com Published by New York Business Global Classification and Enumeration of Primitive Eisenstein Triples using Prime Factorization Techniques Somphong Jitman1, Mohd Sham Mohammad2, Ekkasit Sangwisut3,∗ 1 Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, Thailand 2 Centre for Mathematical Sciences, Universiti Malaysia Pahang Al-Sultan Abdullah, Lebuh Persiaran Tan Khalil Yaakob, Kuantan, Pahang, Malaysia 3 Department of Mathematics and Statistics, Faculty of Science and Digital Innovation, Thaksin University, Phattalung, Thailand Abstract. This study delves into the concept of primitive Eisenstein triples, defined as positive integer solutions (a, b, c) to the quadratic equation a2 − ab + b2 = c2, subject to the condition a < c < b and gcd(a, b, c) = 1. We classify these triples according to the prime factorization of the integer c, elucidating how their existence is intricately linked to specific congruence conditions imposed on the prime divisors of c. Furthermore, we establish a bijective correspondence between these triples and a certain subset of the unit circle. This correspondence enables a comprehensive enumeration of the triples and precisely characterizes the conditions under which such solutions exist. 2020 Mathematics Subject Classifications: 11D09, 11D45, 20K25 Key Words and Phrases: Eisenstein triples, Eisenstein integers, abelian groups, prime number 1. Introduction The study of positive integer solutions to the equation a2 + b2 = c2, known as Pythagorean triples, has fascinated mathematicians for centuries. Equivalently, the in- tegers a, b and c are the lengths of a right-angled triangle. These triples, characterized by positive integers a, b and c satisfying the equation, serve as a cornerstone of number theory and geometry. Beyond their classical role in mathematics, Pythagorean triples have deep connections to algebraic structures, modular forms, and applications in modern cryptography and coding theory. In [1], W. Sierpinski remarked that “It would be more difficult to prove the exis- tence of an arbitrary number of primitive Pythagorean triples with the same hypotenuse.” ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6781 Email addresses: sjitman@gmail.com (S. Jitman), mohadsham@umpsa.edu.my (M. Sham), ekkasit@tsu.ac.th (E. Sangwisut) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 2 of 16 Subsequently, in [2], Ch. L. Shedd demonstrated that there are precisely 64 primitive Pythagorean triples with the hypotenuse c = 2, 576, 450, 045 = 5 · 13 · 17 · 29 · 37 · 41 · 53. Twenty-two years later, this question was revisited by E. J. Eckert in [3], the group struc- ture of the set of primitive Pythagorean triples was investigated. Eckert provided necessary and sufficient conditions for the existence of primitive Pythagorean triples with a given hypotenuse. Alternatively, the set of primitive Pythagorean triples can be identified with the group of rational points on the unit circle. As discussed in [4, 5], this group can be decomposed into a direct sum of the unit group of the group of rational points on the unit circle and a free abelian group. This group-theoretic framework enables both the charac- terization and enumeration of primitive Pythagorean triples with a given hypotenuse (see [5]). In a similar manner, Eisenstein triples (60-degree triples) arise from the analogous equation a2 − ab + b2 = c2. These triples correspond to triangles with sides of integer lengths a, b, and c, where the angle opposite the side of length c is 60◦. A general method for constructing all such triples have been established (see [6]). Previous studies, including [7], [8], and [9], have explored parametric equations and have established relationships between 120-degree triples and 60-degree triples. Fundamental properties of Eisenstein triples have been presented in [10]. This paper focuses on the characterization and enumeration of primitive Eisenstein triples for a fixed hypotenuse. In particular, the one-to-one correspondence between these triples and the ω-rational points (see (3) for the definition) in the second sextant of the unit circle. The paper is organized as follows. In Section 2, the concept and basic properties of (primitive) Eisenstein triples are recalled. In Section 3, the concept of the ω-rational unit circle is introduced together with a link between Eisenstein triples and points on the ω-rational unit circle. The group structure of the ω-rational unit circle is presented in Section 4. Based on this group structure, the characterization and enumeration of primitive Eisenstein triples with a fixed hypotenuse are established in Section 5. The summary is given in Section 6. 2. Eisenstein Triples In this section, some properties and geometric interpretation of Eisenstein triples are recalled in terms of triangles with a 60◦ angle. Key results from [10] concerning the classification of these triples and the constraints on the values of c are presented. In addition, a useful partition of the set of all primitive Eisenstein triples is introduced. An Eisenstein triple is a triple (a, b, c) of positive integers with a < c < b, that satisfies the equation a2 − ab + b2 = c2. (1) The Eisenstein triple is said to be primitive if gcd(a, b, c) = 1. The law of cosines states that for any triangle with sides a, b and c, and θ is the angle θ opposite side c, the following equation holds: a2 − 2ab cos θ + b2 = c2. (2) S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 3 of 16 When θ is 60◦, (2) simplifies the form given in (1), which is the equation of an Eisenstein triple (see Figure 1). B C A ca b 60◦ Figure 1: Triangles containing a 60◦ angle. For examples: • The triples (3, 8, 7) and (5, 8, 7) are primitive Eisenstein triples. • The Eisenstein triples (6, 16, 14) and (10, 16, 14) are not primitive. We note that every non-primitive Eisenstein triple can be expressed as a positive multiple of a primitive one. It is therefore sufficient to focus primarily on the study of primitive Eisenstein triples. The concept of conjugate Eisenstein triples is given as follows. Theorem 1 ([10]). If (a, b, c) is a primitive Eisenstein triple, then (b − a, b, c) is also a primitive Eisenstein triple. A primitive Eisenstein triple (b− a, b, c) is called a conjugate of the primitive Eisen- stein triple (a, b, c) (see Figure 2). A B C D b a b− a bc 60◦ 60◦ Figure 2: An Eisenstein triple and its conjugate The length of the side c of a primitive Eisenstein triple (a, b, c) is subject to the following restriction. Theorem 2 ([10, Theorem 1]). If (a, b, c) is a primitive Eisenstein triple, then c is neither a multiple of 2 nor 3. More generally, the only prime factors of c are primes of the form 6k + 1. S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 4 of 16 Based on Theorem 2, we introduce the following useful partition on the set of primitive Eisenstein triples. Let ET denote the set of all primitive Eisenstein triples. For each positive integer c, let ETc denote the subset of ET containing triples whose side opposite an angle 60◦ is c. From Theorem 2, the integer c in every primitive Eisenstein triple must have only prime factors of the form 6k + 1 for some non-negative integer k. This yields the following partition ET = ⊔ c>1 ETc = ET7 ⊔ET13 ⊔ET19 ⊔ . . . , where the union is taken over all values of c whose prime factors are congruence to 1 modulo 6. 3. From Eisenstein Triples to the ω-Rational Unit Circle This section presents a connection between Eisenstein triples and points on the ω- rational unit circle (see (3) for the definition). Using the unique factorization property of Eisenstein integers, we demonstrate how each primitive Eisenstein triple is mapped to an ω-rational point in the unit circle in the complex plane. This mapping not only highlights the geometric structure of Eisenstein triples but also establishes a link between their algebraic and geometric representations. Eisenstein integers are complex numbers of the form z = a + bω, where a, b ∈ Z, and ω = e 2πi 3 = −1+ √ 3i 2 is a primitive cube root of unity. We note that ω and ω are roots of the polynomial x2 + x + 1, where ω is the complex conjugate of ω. It is easily seen that ω3 = ω3 = 1, ω2 = ω = −1 − ω, ω · ω = 1 and ω + ω = −1. The set of all Eisenstein integers, denoted Z[ω], forms a commutative ring with identity under the usual addition and multiplication of complex numbers. The unit group of Z[ω] is U = { 1, ω, ω2,−1,−ω,−ω2 } = ⟨−ω⟩, which is isomorphic to the cyclic group of order 6. The norm of an Eisenstein integer z = a + bω is defined by N(z) = √ zz = √ (a + bω)(a + bω) = √ a2 − ab + b2. For two Eisenstein integers z and z′, the norm satisfies N(zz′) = N(z)N(z′). Two Eisen- stein integers z and z′ are said to be associates if z = δ · z′ for some unit δ ∈ U . The ring Z[ω] is a unique factorization domain (UFD), meaning every nonzero, non-unit element has a unique factorization into a product of irreducible elements, up to the rearrangement of the factors and the replacement of any irreducible element with one of its associates. For more information on Eisenstein integers, the reader may refer to [11–13] and [14]. Let G(R) denote the unit circle in the complex plane. Since N(z) equals the Euclidean norm of z for all complex numbers z, the elements in G(R) can be viewed as their ω- expansions. Precisely, G(R) := { ζ = u + vω ∈ C ∣∣∣u, v ∈ R, N(ζ) = √ u2 − uv + v2 = 1 } . S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 5 of 16 The set G(R) forms an abelian group under the standard multiplication in C. Moreover, N(1) = 1, N(ζ1ζ2) = N(ζ1)N(ζ2), and N(ζ−1) = N(ζ)−1. Let G(Q) be the subset of G(R) of the form G(Q) := { ζ = u + vω ∣∣∣u, v ∈ Q, N(ζ) = √ u2 − uv + v2 = 1 } . (3) The set G(Q) is called the ω-rational unit circle and each element in G(Q) is called an ω-rational point. We observe that although i ∈ G(R) but it is not in G(Q). It is interesting to investigate properties further. Proposition 1. The set G(Q) is a subgroup of G(R). Proof. It is easy to see that 1 = 1 + 0 · ω ∈ G(Q) which implies that G(Q) ̸= ∅. For elements u1 + v1ω, u2 + v2ω ∈ G(Q), the product (u1 + v1ω)(u2 + v2ω)−1 = u1+v1ω u2+v2ω can be expressed as u1+v1ω u2+v2ω = (u1+v1ω)(u2+v2ω) u2 2−u2v2+v22 . Simplifying the numerator and denominator yields (u1u2+v1v2−u1v2)+(v1u2−u1v2)ω u2 2−u2v2+v22 . This can be written as u1u2+v1v2−u1v2 u2 2−u2v2+v22 + v1u2−u1v2 u2 2−u2v2+v22 ω. Since u1, v1, u2, v2 ∈ Q, the result is in G(Q). For a given primitive Eisenstein triple (a, b, c), define the associated Eisenstein integer z := a + bω, whose norm is given by N(z) = √ a2 − ab + b2 = c. It follows immediately that 2a ̸= b. Otherwise, we would have c = √ 3a, which contradicts the assumption that c is an integer. The corresponding point in the unit circle is the normalized complex number ζ := z N(z) = a c + b c ω = 2a− b 2c + b √ 3 2c i ∈ G(R). Since a c and b c are rational numbers, ζ is an ω-rational point in G(Q). Consequently, each primitive Eisenstein triple in ET gives rise to an ω-rational point in the ω-rational unit circle G(Q) illustrated in Figure 3. Next, we show that ζ lies in the second sextant. Since Im(ζ) > 0, it follows that ζ lies in the upper half plane. We note that the slope of the line from the origin to ζ is m = b √ 3 2a− b which implies that m > √ 3 if 2a > b, or m < − √ 3 if 2a < b. In both cases, ζ lies in the second sextant. Let ζ := u + vω ∈ G(Q) ∖ {ω,−ω} be in the second sextant. Let d denote the least common multiple of the denominators of u and v. Then dζ = du + dvω ∈ Z[ω] which ensures that the coordinates of the associated Eisenstein triple (du, dv, d) are integral. Alternatively, we may consider the primitive representation of the triple (du, dv, d) by dividing by the greatest common divisor. S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 6 of 16 Figure 3: Geometric projection of the Eisenstein integer a+ bω into the ω-rational unit circle by normalization. For example, take ζ = 18 42 + 120 105ω ∈ G(Q). Then the least common multiple of the denominators is d = lcm(42, 105) = 210 which implies that 210ζ = 90 + 240ω ∈ Z[ω]. Hence, the corresponding Eisenstein triple is (90, 240, 210) whose primitive representative (after dividing by the greatest common divisor 30) is (3, 8, 7) ∈ ET. The set of associate elements of ζ ∈ G(Q) is denoted by Uζ = { ζ, (−ω)ζ, (−ω)2ζ, (−ω)3ζ, (−ω)4ζ, (−ω)5ζ } (4) which represents six distinct points on G(Q), each located in a different sextant (see Figure 4). In other words, multiplying ζ by −ω results in a 60◦ clockwise rotation of ζ. Figure 4: The associate elements of ζ in G(Q). S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 7 of 16 Let ζ be a point in G(Q)∖U . Let f br a function responsible for rotating ζ clockwise into the second sextant. Precisely, f(ζ) = (−ω)i−2ζ for all ζ in the ith sextant (for 1 ≤ i ≤ 6). This operation ensures that f(ζ) locates in the second sextant. Next, let g be a function that maps points on the second sextant to the set of Eisenstein triples ET. Finally, let et = g ◦ f (5) be a composite function that maps any point ζ ∈ G(Q) ∖ U to ET by first rotating ζ clockwise into the second sextant and then applying g. We demonstrate the infinitude of the set of primitive Eisenstein triples by showing that G(Q) contains infinitely many points. Proposition 2. The set of primitive Eisenstein triples is infinite. Proof. From (5), it is not difficult to see that the function et : G(Q) ∖ U → ET is surjective and it is a 6-to-1 map. To complete the proof, it suffices to show that G(Q) has infinite order. For each element m ∈ Q, let zm = 1−2m 1−m+m2 + 1−m2 1−m+m2ω. Then 1−2m 1−m+m2 , 1−m2 1−m+m2 ∈ Q and N(zm) = √( 1−2m 1−m+m2 )2 − ( 1−2m 1−m+m2 )( 1−m2 1−m+m2 ) + ( 1−m2 1−m+m2 )2 = 1. It follows that zm ∈ G(Q) for all m ∈ Q. Since zm ̸= zn for all m ̸= n ∈ Q, G(Q) contains infinitely many ω-rational points. Hence, G(Q) has infinite order. Table 1 illustrates a behavior of the element ζ = 3 7 + 8 7ω in G(Q) and their successive powers ζn for all integers −3 ≤ n ≤ 3. • The corresponding values of ζn in the second column. • The associated Eisenstein triples et(ζn) = (an, bn, cn) in the third column. 4. Prime Numbers and the Abelian Group Structure This section examines the structure of prime numbers in the ring Z[ω], focusing on their factorization and classification based on residue classes modulo 3. Based on the unique factorization in this ring, it can be shown that the group G(Q) can be decomposed as a direct sum of the unit group and a free abelian group. S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 8 of 16 n ζn et(ζn) = (an, bn, cn) −3 323 343 + 360 343ω (323, 360, 343) −2 −39 49 + 16 49ω (16, 55, 49) −1 −5 7 − 8 7ω (5, 8, 7) 1 3 7 + 8 7ω (3, 8, 7) 2 −55 49 − 16 49ω (39, 55, 49) 3 − 37 343 − 360 343ω (37, 360, 343) Table 1: Powers of ζ = 3 7 + 8 7 ω in G(Q) and their associated primitive Eisenstein triples. Let P denote the set of positive prime numbers. For i ∈ {1, 2, 3}, let Pi denote the set of prime numbers congruence i modulo 3. Then P can be partitioned into residue classes modulo 3 as follows: P = P1 ⊔ P2 ⊔ P3 = {7, 13, 19, 31, . . . } ⊔ {2, 5, 11, 17, . . . } ⊔ {3}. (6) While P3 is a singleton, it can be seen that P1 and P2 are infinite sets, For each p ∈ P1, we have p ≡ 1 (mod 3) which can be expressed as p = a2 − ab + b2 for some integers a and b (see [15]). This allows us to express the factorization p = (a + bω)(a + bω) (see [11, 12]). Let q := a + bω. Then the conjugate of q is q = a + bω. Let ζp be the complex number of the form ζp = q q = a + bω a + bω . (7) Since ζp has rational coordinates and its norm satisfies N(ζp) = N(a+bω) N(a+bω) = √ a2−ab+b2√ a2−ab+b2 = 1, it follows that ζp belongs to the group G(Q). Example 1. Let p = 7. Then 7 = (1 + 3ω) · (1 + 3ω) = q · q, where q = 1 + 3ω and q = 1 + 3ω. It follows that ζ7 = q q = 1 + 3ω 1 + 3ω is an element in G(Q). Similarly, for p = 13, we write 13 = (1 + 4ω)(1 + 4ω), where q = 1 + 4ω and q = 1 + 4ω. Hence, ζ13 = 1 + 4ω 1 + 4ω ∈ G(Q). We note that Z[ω] is a unique factorization domain with the unit group U = { 1, ω, ω2, −1,−ω,−ω2 } . The classification of the irreducible elements in Z[ω] is recalled as follows. For more information on irreducible elements in Z[ω], the reader may refer to [14] and [12, p. 110]. Based on the partition in (6) of P , there are of three types of partitions. • For each p ∈ P1, the irreducible factorization of p in Z[ω] is given by p = (a + bω)(a + bω), where a + bω and a + bω are not associated. In this case, we have U(a + bω) ̸= U(a + bω) and N(a + bω) = N(a + bω) = √ p. S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 9 of 16 • For each p ∈ P2, p is an irreducible element in Z[ω] and N(p) = p. • For the positive prime integer p = 3, its factorization is 3 = (1 + 2ω)(1 + 2ω) = −(1 + 2ω)2. In this case, 1 + 2ω and 1 + 2ω are associated and 1 + 2ω is irreducible in Z[ω]. Furthermore, N(1 + 2ω) = √ 3. As the ring Z[ω] contains the ring Z, this implies that prime integers in P1 and P3 can be factored further. Define an equivalence relation on the set Z[ω] by z ∼ z′ if and only if z and z′ are associated. Each equivalence class under the relation ∼ consists of all elements in Z[ω] that are associated elements of z. Precisely, the equivalence class containing z is Uz = { z, (−ω)z, (−ω)2z, (−ω)3z, (−ω)4z, (−ω)5z } , which represents points on the circle of radius N(z) in different sextants (see Figure 4). Consequently, the partition of P in (6) is defined as follows. Let Q denote the set of irreducible elements in the ring Z[ω]. The Q has a partition of the form Q := Q1 ⊔Q1 ⊔Q2 ⊔Q3 (8) with the following conditions. (i) For each p ∈ P1, if the factorization of p in Z[ω] is p = q · q, the elements q and q are assigned to the sets Q1 and Q1, respectively. (ii) Q2 := P2. (iii) Q3 := {1 + 2ω}. It follows that every nonzero element z ∈ Z[ω] can be uniquely expressed in the form z = δ · k∏ i=1 qeii , (9) where δ ∈ U , k ≥ 0, qi is an irreducible element in Q, and ei ≥ 1 is an integer. This factorization is unique up to the rearangement of the factors and the replacement of any irreducible element with one of its associates. Next, the decomposition of G(Q) will be presented using the field of fractions of Z[ω]. Let Q[ω] = {u + vω | u, v ∈ Q} ⊆ C. Then Q[ω] is the field of fractions of Z[ω]. Analogous to (9), each element z ∈ Q[ω] ∖ {0} can be uniquely expressed in the form z = δ · k∏ i=1 qeii , (10) where δ ∈ U , k ≥ 0, qi is an irreducible element in Q, and ei ∈ Z ∖ {0}. The following theorem describes the decomposition of G(Q). S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 10 of 16 Theorem 3. The abelian group G(Q) is decomposed as an internal direct sum: G(Q) = U ⊕ F, where U = { 1, ω, ω2,−1,−ω,−ω2 } is the unit group of Z[ω], and F is a free abelian group with basis given by the collection {ζp | p ∈ P1}, where P1 is the set of prime integers congruent to 1 (mod 3). Proof. Let z ∈ G(Q). Then z ∈ Q[ω] which has a unique factorization in the form of (10). Rearrange the factorization using the set of irreducible elements Q in (8), we have z = δ · ( r∏ i=1 qcii q di i ) · ( s∏ i=1 q̃j ej ) · (1 + 2ω)t, (11) where δ ∈ U, q1, . . . , qr ∈ Q1, q1, . . . , qr ∈ Q1, q̃1, . . . , q̃s ∈ Q2, and the exponents c1, . . . , cr, d1, . . . , dr, e1, . . . , es and t are integers. Since z is an ω-rational point in the ω-rational unit circle, we have N(z) = 1. Expanding N(z) using its factorization, it follows that 1 = N(z) = N(δ) ·N ( r∏ i=1 qcii q di i ) ·N ( s∏ i=1 q̃j ej ) ·N(1 + 2ω)t = ( r∏ i=1 N(qi) ciN(qi) di )( s∏ i=1 N (q̃j) ej ) · (√ 3 )t = ( r∏ i=1 N(qi) ciN(qi) di )( s∏ i=1 N (q̃j) ej ) · (√ 3 )t . We note that N(qi) = N(qi) for qi ∈ Q1 and qi ∈ Q1. Since N(z) = 1, we deduce that ci = −di, and ej and t are zero. Substituting these constrains into (11), it can be concluded that z = δ · ( r∏ i=1 qcii q −ci i ) = δ · r∏ i=1 ( qi qi )ci . By setting ζi = qi qi as in (7), we further deduced that z = δ · r∏ i=1 ζcii ∈ U ⊕ F. This completes the proof. S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 11 of 16 5. Characterization and Enumeration of Primitive Eisenstein Triples In this section, we focus on the characterization and enumeration of primitive Eisen- stein triples. We begin with a formula for the hypotenuse c of an Eisenstein triple. Next, we present necessary and sufficient conditions for the existence of primitive Eisenstein triples with a given hypotenuse c, highlighting the relationship between the prime factor- ization of c and the structure of the corresponding triples. This allows us to establish the enumeration of primitive Eisenstein triples with a given hypotenuse c. Finally, we pro- vide examples that demonstrate how these results can be applied to specific values of c, illustrating the number of primitive Eisenstein triples associated with particular integers. Lemma 1. For a positive integer k, distinct primes p1, . . . , pk in P1, nonzero integers n1, . . . , nk, and signs ϵ1, . . . , ϵk ∈ {±1}, define (a, b, c) := et ( k∏ i=1 ζϵi·ni pi ) ∈ ET . Then c = pn1 1 . . . pnk k . Proof. Let ζ = ∏k i=1 ζ ϵi·ni pi . By the defining ζpi = qi qi given in (7), it can be rewritten in the form of ζ = k∏ i=1 ζϵi·ni pi = k∏ i=1 ( qi qi )ϵi·ni . Since p1, . . . , pk are in P1, we have the factorization pi = qi ·qi over Z[ω] for all i = 1, . . . , k. It follows that c = k∏ i=1 pni i = k∏ i=1 (qi · qi) ni . Let z := ζ · c = k∏ i=1 ( qi qi )ϵi·ni · (qi · qi) ni = k∏ i=1 q (ϵi+1)·ni i q (ϵi−1)·ni i = k∏ i=1 q̃i 2ni , where q̃i is defined to be q̃i = { qi if ϵi = 1, qi if ϵi = −1. Hence, z ∈ Z[ω] and its norm satisfies N(z) = N (∏k i=1 q̃i 2ni ) = ∏k i=1 p ni i = c. We can choose δ ∈ U such that the associate δz = a + bω is in the second sextant. Hence, (a, b, c) forms an Eisenstein triple. Next, we show that the Eisenstein triple (a, b, c) is primitive. For contrary, we suppose that there exists pi such that pi | a and pi | b. Then a = pia ′ and b = pib ′ for some integers a′ and b′. Substituting these expressions into δz, we obtain δz = a + bω = pia ′ + pib ′ω = pi (a′ + b′ω) which implies that pi | z. Since z = ∏k i=1 q̃i 2ni and pi = qi ·qi, qi and qi appear S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 12 of 16 as factors in z. This is a contradiction. Consequently, there a and b has no common prime divisors. As a result, the Eisenstein triple (a, b, c) must be primitive. The following theorem provides a classification of primitive Eisenstein triples based on the prime factorization of the hypotenuse c. Theorem 4. Let c > 1 be an integer with prime factorization c = pn1 1 . . . pnk k , where k is a positive integer, p1, . . . , pk are distinct prime numbers, and n1, . . . , nk are positive exponents. Then exactly one of the following statements holds: (i) If pi ≡ 1 (mod 3) for all i = 1, . . . , k, then the map et|Z : Z → ETc is a bijection, where Z = {∏k i=1 ζ ϵi·ni pi | ϵ1, . . . , ϵk ∈ {±1} } , and ζpi is a complex number defined in (7) for the prime pi. (ii) Otherwise, the set ETc is empty. Proof. To prove 1), assume the notations as in Section 3. Consider the surjective function et : G(Q) ∖ U → ET defined in (5). Let Ω be a relation on G(Q) ∖ U given by ζ1Ωζ2 if and only if ζ−1 1 ζ2 ∈ U . Then Ω is an equivalence relation and we denote the corresponding quotient set by (G(Q) ∖ U)/Ω. Consequently, the induced function et : (G(Q) ∖ U)/Ω → ET is bicjective. By the Theorem 3, we have G(Q) = U ⊕ F , which implies that G(Q) ∖ U = (U ⊕ F ) ∖ U = U ⊕ (F ∖ {1}). Since every element in U ⊕ F is uniquely of the form uf with u ∈ U and f ∈ F , the elements not in U are exactly those with f ̸= 1. Hence, (U ⊕ F ) \ U = {uf | u ∈ U, f ∈ F \ {1}} = U ⊕ (F \ {1}). Thus, the quotient sets are identical: (G(Q) ∖ U)/Ω = (U ⊕ (F ∖ {1}))/Ω = F ∖ {1}. This establishes the one-to-one correspondence et : F ∖ {1} → ET . Now, let p1, . . . , pk be distinct primes in P1 and let n1, . . . , nk be positive integers. Let Z := { k∏ i=1 ζϵi·ni pi | ϵ1, . . . , ϵk ∈ {±1} } . S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 13 of 16 Then Z ⊆ F ∖ {1} and the restriction map et |Z : Z → ET is injective, and the image of Z is ETc (see Lemma 1). Therefore, et |Z : Z → ETc is a bijection. From Theorem 2, 2) follows immediately. Corollary 1. Let c > 1 be an integer with prime factorization as described in Theorem 4. Then, exactly one of the following holds: (i) If pi ≡ 1 (mod 6) for all i = 1, . . . , k, then there are 2k primitive Eisenstein triples with hypotenuse c. (ii) Otherwise, there are no primitive Eisenstein triples with hypotenuse c. Proof. The first statement follows form the one-to-one correspondence established in 1) of Theorem 4. Precisely, |ETc| = ∣∣∣∣∣ { k∏ i=1 ζϵi·ni pi | ϵ1, . . . , ϵk ∈ {±1} }∣∣∣∣∣ = 2k. The second statement can be deduced directly from 2) of Theorem 4. The following examples illustrate the application of Theorem 4 and Corollary 1 to determine the primitive Eisenstein triples for specific hypotenuses. Example 2. Let c = 49. The prime factorization is c = 72 which 7 ∈ P1. By Example 1, we have ζ7 = 1+3ω 1+3ω = −8 7 − 3 7ω and ζ−1 7 = 1+3ω 1+3ω = −5 7 + 3 7ω. We have the set Z = { ζϵ·27 | ϵ ∈ {±1} } = { ζ27 , ζ −2 7 } . By Theorem 4, there are two cases: Case ϵ1 = 1: We compute ζ27 : ζ27 = ( −8 7 − 3 7 ω )2 = 55 49 + 39 49 ω. This lies in the sixth sextant. Its associate in the second sextant is 16 49 + 55 49ω. The corre- sponding primitive Eisenstein triples is et ( 16 49 + 55 49ω ) = (16, 55, 49). Case ϵ1 = −1: We compute ζ−2 7 : ζ−2 7 = ( −5 7 + 3 7 ω )2 = 16 49 − 39 49 ω. This lies in the fifth sextant. Its associate in the second sextant is 39 49 + 55 49ω. The corre- sponding primitive Eisenstein triples is et ( 39 49 + 55 49ω ) = (39, 55, 49). Consequently, ET49 = {(16, 55, 49), (39, 55, 49)}. S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 14 of 16 Example 3. Let c = 91. Then the prime factorization is c = 7 · 13, where 7, 13 ∈ P1. From Example 1, we have ζ7 = 1 + 3ω 1 + 3ω , and ζ13 = 1 + 4ω 1 + 4ω . Then Z = {ζϵ17 ζϵ213 | ϵ1, ϵ2 ∈ {±}} = {ζ7ζ13, ζ−1 7 ζ13, ζ7ζ −1 13 , ζ−1 7 ζ−1 13 }. By Theorem 4, we analyze the following four cases: Case 1: ϵ1 = 1, ϵ2 = 1. Compute ζ7 · ζ13: ζ7 · ζ13 = 1 + 3ω 1 + 3ω · 1 + 4ω 1 + 4ω = 96 91 + 85 91 ω. This lies in the sixth sextant. Its associate in the second sextant is 11 91 + 96 91ω. The corre- sponding primitive Eisenstein triples is et ( 11 91 + 96 91ω ) = (11, 96, 91). Case 2: ϵ1 = −1, ϵ2 = 1. Compute ζ−1 7 · ζ13: ζ−1 7 · ζ13 = 1 + 3ω 1 + 3ω · 1 + 4ω 1 + 4ω = 99 91 + 19 91 ω. This lies in the sixth sextant. Its associate in the second sextant is 80 91 + 99 91ω. The corre- sponding primitive Eisenstein triples is et ( 80 91 + 99 91ω ) = (80, 99, 91). Case 3: ϵ1 = 1, ϵ2 = −1. Compute ζ7 · ζ13: ζ7 · ζ−1 13 = 1 + 3ω 1 + 3ω · 1 + 4ω 1 + 4ω = 80 91 − 19 91 ω. This lies in the fifth sextant. Its associate in the second sextant is 19 91 + 99 91ω. The corre- sponding primitive Eisenstein triples is et ( 19 91 + 99 91ω ) = (19, 99, 91). Case 4: ϵ1 = −1, ϵ2 = −1. Compute ζ7 · ζ13: ζ−1 7 · ζ−1 13 = 1 + 3ω 1 + 3ω · 1 + 4ω 1 + 4ω = 11 91 − 85 91 ω. This lies in the fifth sextant. Its associate in the second sextant is 85 91 + 96 91ω. The corre- sponding primitive Eisenstein triples is et ( 85 91 + 96 91ω ) = (85, 96, 91). Therefore, ET49 = {(11, 96, 91), (85, 96, 91), (80, 99, 91), (19, 99, 91)}. 6. Conclusion This paper develops a unified framework for primitive Eisenstein triples, those integer triangles with a 60–degree angle and side lengths satisfying the classical Eisenstein relation. The key idea is to translate triples into points on the ω–rational unit circle and back. Using unique factorization in the ring of Eisenstein integers, we show that the ω–rational unit circle splits cleanly into two parts: a finite set of six units and a free abelian group with basis given by the collection {ζp | p ∈ P1}, where P1 is the set of primes congruent to one modulo three. Every ω–rational point can be written uniquely as a unit times a product of S. Jitman, M. Mohammad, E. Sangwisut / Eur. J. Pure Appl. Math, 18 (4) (2025), 6781 15 of 16 these generators. Choosing a canonical representative in the second sextant and clearing denominators gives a direct and lossless way to pass from points to primitive triples. This perspective yields sharp existence and counting results for a fixed hypotenuse. Write the hypotenuse as a product of rational primes. Primitive Eisenstein triples exist exactly when every prime factor is congruent to one modulo three. In that case, if there are k distinct such primes, the number of primitive triples with that hypotenuse is exactly two to the power k. Practically, the test for existence reduces to a single inspection of the prime factorization, and enumeration follows immediately. Algorithmically, recovering the side lengths from a chosen omega–rational point—or directly from the factorization of the hypotenuse—enables efficient generation at scale. It would be interesting to investigate whether the distribution and averaging of counts for a fixed hypotenuse admit a generating-function treatment in which Stirling numbers of the second kind or Whitney numbers arise naturally; we leave this as future work (see, e.g., [16, 17]). Acknowledgements S. 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