A Ring of Pythagorean Triples over Quadratic Fields
Let K be a quadratic field and let R be the ring of integers of K such that R is a unique factorization domain. The set P of all Pythagorean triples in R is partitioned into P η , sets of triples 〈α, β, γ〉 in P where η = γ − β. We show the ring structures of each P η and P from the ring structur...
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| author | Harnchoowong, A. Somboonkulavudi, C. Гарнчоошонг, А. Сомбоонкулавуді, Ц. |
| author_facet | Harnchoowong, A. Somboonkulavudi, C. Гарнчоошонг, А. Сомбоонкулавуді, Ц. |
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| description | Let K be a quadratic field and let R be the ring of integers of K such that R is a unique factorization domain. The set P of all Pythagorean triples in R is partitioned into P η , sets of triples 〈α, β, γ〉 in P where η = γ − β. We show the ring structures of each P η and P from the ring structure of R. |
| first_indexed | 2026-03-24T02:19:03Z |
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| fulltext |
UDC 512.5
C. Somboonkulavudi, A. Harnchoowong (Chulalongkorn Univ., Bangkok, Thailand)
A RING OF PYTHAGOREAN TRIPLES OVER QUADRATIC FIELDS*
КIЛЬЦЕ ПIФАГОРОВИХ ТРIЙОК НАД КВАДРАТНИМИ ПОЛЯМИ
Let K be a quadratic field and let be R the ring of integers of K such that R is a unique factorization domain. The set P
of all Pythagorean triples in R is partitioned into Pη, sets of triples 〈α, β, γ〉 in P where η = γ − β. This paper shows the
ring structures of each Pη and P from the ring structure of R.
Нехай K — квадратне поле, а R — кiльце цiлих з K таких, що R — єдина факторизацiйна область. Множина P всiх
пiфагорових трiйок з R разбивається на Pη, множини трiйок 〈α, β, γ〉 в P, де η = γ−β. В роботi показано кiльцевi
структури для кожного Pη та P з кiльцевої структури R.
1. Introduction. A triple 〈α, β, γ〉 of elements of a ring is said to be a Pythagorean triple if
α2 + β2 = γ2. B. Dawson [1] defined operations on the set of all Pythagorean triples in Z so that
this set is a ring. J. T. Cross [2] displayed a method for generating all Pythagorean triples over the
ring of Gaussian integers.
Let K be a quadratic extension of Q such that the ring of integers R of K is a unique factorization
domain. Let P be the set of all Pythagorean triples in R, i.e.,
P = {〈α, β, γ〉 ∈ R3 | α2 + β2 = γ2}.
The set P is partitioned into sets
Pη = {〈α, β, γ〉 ∈ P | γ − β = η}
for all η ∈ R. This paper shows how to find all elements of each Pη with all elements of P as the
byproducts and define bijections between Pη and R, which construct a one-to-one correspondence
between P and R×R.
2. Preliminaries. Throughout this paper, all variables will be assumed to represent algebraic
integers unless otherwise stated. The notation dre will be used for the smallest rational integer
greater than or equal to the real number r.
The parity is significant in many theorems about Pythagorean triples. James T. Cross shows that
δ := 1 + i plays a role in the ring of Gaussian integers like that played by 2 in Z [2]. We expand his
idea by using the following theorem.
Theorem 2.1. Let K = Q(
√
d), where d is a squarefree integer, R be the ring of integers of K.
Then:
2 is ramified in R if d ≡ 2 or 3 (mod 4).
2 splits completely in R if d ≡ 1 (mod 8).
2 is inert in R if d ≡ 5 (mod 8).
We will separate each case of R into three sections. If 2 is ramified in R, there is a prime δ ∈ R
such that 2 ∼ δ2 and |R/〈δ〉| = 2. For α ∈ R, we may say that α is even if α is divisible by δ
and α is odd otherwise. Moreover, the sum of two even or two odd algebraic integers gives an even
* The first author was supported by Chulalongkorn University Graduate Scholarship to Commemorate the 72nd Anni-
versary of His Majesty King Bhumibol Adulyadej.
c© C. SOMBOONKULAVUDI, A. HARNCHOOWONG, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1 135
136 C. SOMBOONKULAVUDI, A. HARNCHOOWONG
one, the sum of an even algebraic integer and an odd one gives an odd one, the product of two odd
ones gives an odd one, and the product of an even one and any algebraic integer gives an even one.
Furthermore, since δ | 2, 2 and all integers that are divisible by 2 are even algebraic integers. All units
in R are odd. In case that 2 splits completely in R, there are non-associate primes δ, δ ∈ R such that
2 ∼ δδ and |R/〈δ〉| =
∣∣R/〈δ〉∣∣ = 2. If 2 is inert in R, 2 is a prime in R.
Let π be a prime in R. The set R \ πR contains all elements of R which are not divisible by π.
We use the countability property of R to show a connection between R \ πR and R which leads to
a one-to-one correspondence between Pη and R.
Definition 2.1. Let π be a prime in R. All non-associate primes in R can be put into order, say
π, π1, π2, π3, . . . . Define Ψπ : (R \ πR)→ R by
Ψπ(uπa11 π
a2
2 π
a3
3 . . .) = uπa1πa21 π
a3
2 . . . ,
where {a1, a2, . . .} ⊂ Z+
0 and u is a unit in R. It is not difficult to see that the mapping Ψπ is a
one-to-one correspondence.
For the case that η = 0, P0 = {〈0, β, β〉 |β ∈ R} and the mapping ϕ : P0 → R defined by
ϕ(〈0, β, β〉) = β is a one-to-one correspondence. The following theorems consider the case where
η 6= 0.
3. 2 is ramified in R. In this case, there is a prime δ ∈ R such that 2 ∼ δ2 and |R/〈δ〉| = 2.
To show a ring structure of Pη and P, we characterize Pη and define bijections by considering two
cases of η where δa0 ||η for a0 = 0, 1 and δ2 | η in the following theorems.
Theorem 3.1. Let η be an algebraic integer and η = uδa0πa11 π
a2
2 . . . πamm , where a0 = 0, 1 and
for k ≥ 1, ak ∈ Z+
0 , u is a unit and πk ∈ R are non-associate odd primes. Set ρ = δa0πb11 π
b2
2 . . . πbmm ,
where bk =
⌈ak
2
⌉
. Then Pη is{〈
α,
α2 − η2
2η
,
α2 + η2
2η
〉 ∣∣∣∣ α = τρ for some odd τ ∈ R
}
.
Moreover, the mapping ϕ : Pη → R defined by
ϕ(〈α, β, γ〉) = Ψδ
(
α
ρ
)
is a one-to-one correspondence.
Proof. Suppose 〈α, β, γ〉 ∈ Pη. Since η = γ − β, we have 〈α, β, γ〉 = 〈α, (α2 − η2)/2η,
(α2 + η2)/2η〉. Then 2η |α2+η2 and thus δa0+2πa11 π
a2
2 . . . πamm |α2+u2δ2a0π2a11 π2a22 . . . π2amm . Hence
δ2a0πa11 π
a2
2 . . . πamm |α2. Since for each k = 1, . . . ,m, bk =
⌈ak
2
⌉
, we get δa0πb11 π
b2
2 . . . πbmm |α.
Therefore, there exist an algebraic integer τ such that α = τρ. If τ is even, then δa0+2 |α2 and thus
δa0+2 | δ2a0π2a11 π2a22 . . . π2amm . This is a contradiction, so τ is odd.
Conversely, suppose α = τρ where τ is odd. We have α2 − η2 = τ2δ2a0π2b11 π2b22 . . . π2bmm −
− u2δ2a0π2a11 π2a22 . . . π2amm = δ2a0(τ2π2b11 π2b22 . . . π2bmm − u2π2a11 π2a22 . . . π2amm ). Since 2bk ≥ ak, we
obtain πa11 π
a2
2 . . . πamm |α2 − η2. If a0 = 0, i.e., η and ρ are odd, α + η and α − η are divisible
by δ. Then δ2|α2 − η2. If a0 = 1, since τ2π2b11 π2b22 . . . π2bmm and u2π2a11 π2a22 . . . π2amm are odd and
the difference of these two numbers is even, δa0+2 |α2 − η2. Consequently, 2η |α2 − η2 and thus
2η |α2 + η2.
If 〈α, β, γ〉 ∈ Pη, then α/ρ is an odd algebraic integer and Ψδ(α/ρ) makes the mapping ϕ
injective and surjective.
Theorem 3.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
A RING OF PYTHAGOREAN TRIPLES OVER QUADRATIC FIELDS 137
Theorem 3.2. Let η be an even algebraic integer and η = uδa0πa11 π
a2
2 . . . πamm , where a0 ≥
≥ 2 and for k ≥ 1, ak ∈ Z+
0 , u is a unit and πk ∈ R are non-associate odd primes. Set ρ =
= δb0πb11 π
b2
2 . . . πbmm , where b0 =
⌈
a0 + 2
2
⌉
and bk =
⌈ak
2
⌉
. Then Pη is
{〈
α,
α2 − η2
2η
,
α2 + η2
2η
〉 ∣∣∣∣ α = τρ for some τ ∈ R
}
.
Moreover, the mapping ϕ : Pη → R defined by
ϕ(〈α, β, γ〉) =
α
ρ
is a one-to-one correspondence.
Proof. Suppose 〈α, (α2 − η2)/2η, (α2 + η2)/2η〉 ∈ Pη. Then δa0+2πa11 π
a2
2 . . . πamm |α2 +
+u2δ2a0π2a11 π2a22 . . . π2amm . Therefore, δa0+2πa11 π
a2
2 . . . πamm |α2. Hence δb0πb11 π
b2
2 . . . πbmm |α. Thus
α = τρ for some τ ∈ R.
Conversely, suppose α = τρ, where τ ∈ R. We have α2 = τ2δ2b0π2b11 π2b22 . . . π2bmm which is
divisible by 2η. Moreover, η2 is divisible by 2η because 2 | η. Hence 2η |α2 + η2.
Since any algebraic integer can be written in the form α/ρ, the mapping ϕ is bijective.
Theorem 3.2 is proved.
4. 2 splits completely in R. There are non-associate primes δ, δ ∈ R such that 2 ∼ δδ and
|R/〈δ〉| =
∣∣R/〈δ〉∣∣ = 2. Notice that the ideas of even and odd we used in the proofs of the previous
theorems are also practical in this section where we consider three cases of η depending on the
divisibility by δ and δ. Note that δ and δ hold the same properties and can be switched around in the
following theorem.
Theorem 4.1. Let η ∈ R and η = uδ
a0
πa11 . . . πamm , where a0 ≥ 1, and for k ≥ 1, ak ∈ Z+
0 ,
u is a unit and πk ∈ R are non-associate primes where πk � δ, δ. Set ρ = δ
b0
πb11 . . . πbmm , where
b0 =
⌈
a0 + 1
2
⌉
and bk =
⌈ak
2
⌉
. Then Pη is
{〈
α,
α2 − η2
2η
,
α2 + η2
2η
〉 ∣∣∣∣ α = τρ for some τ ∈ R, where δ - τ
}
.
Moreover, the mapping ϕ : Pη → R defined by
ϕ(〈α, β, γ〉) = Ψδ
(
α
ρ
)
is a one-to-one correspondence.
Proof. Suppose 〈α, (α2 − η2)/2η, (α2 + η2)/2η〉 ∈ Pη. Then 2η |α2 + η2 and thus
δδ
a0+1
πa11 π
a2
2 . . . πamm |α2 + u2δ
2a0
π2a11 π2a22 . . . π2amm . Therefore, δ
a0+1
πa11 π
a2
2 . . . πamm |α2. Hence
δ
b0
πb11 π
b2
2 . . . πbmm |α. Thus α = τρ for some τ ∈ R. If δ|τ, then δ |α2 and δ |u2δ2a0π2a11 π2a22 . . . π2amm ,
a contradiction. This means that δ - τ.
Conversely, suppose α = τρ where τ ∈ R and δ - τ. We have δη |α2 + η2. Since δ - α2
(odd wrt δ) and δ - η2, we have δ |α2 + η2 (even wrt δ). Since 2 ∼ δδ, 2η |α2 + η2.
Theorem 4.1 is proved.
The proofs of the next two theorems are left to the reader.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
138 C. SOMBOONKULAVUDI, A. HARNCHOOWONG
Theorem 4.2. Let η ∈ R and η = uδa0δ
a0
πa11 . . . πamm , where a0 ≥ 1, a0 ≥ 1, and for
k ≥ 1, ak ∈ Z+
0 , u is a unit and πk ∈ R are non-associate primes, where πk � δ, δ. Set
ρ = δb0δ
b0
πb11 . . . πbmm , where b0 =
⌈
a0 + 1
2
⌉
, b0 =
⌈
a0 + 1
2
⌉
and bk =
⌈ak
2
⌉
. Then Pη is{〈
α,
α2 − η2
2η
,
α2 + η2
2η
〉 ∣∣∣∣ α = τρ for some τ ∈ R
}
.
Moreover, the mapping ϕ : Pη → R defined by
ϕ(〈α, β, γ〉) =
α
ρ
is a one-to-one correspondence.
The following theorem uses the idea that all non-associate primes in R can be put into order, say
δ, δ, π1, π2, . . . .
Theorem 4.3. Let η ∈ R and η = uπa11 . . . πamm , where for k ≥ 1, ak ∈ Z+
0 , u is a unit and
πk ∈ R are non-associate primes where πk � δ, δ. Set ρ = πb11 . . . πbmm , where bk =
⌈ak
2
⌉
. Then Pη
is {〈
α,
α2 − η2
2η
,
α2 + η2
2η
〉 ∣∣∣∣ α = τρ for some τ ∈ R, where δ - τ, δ - τ
}
.
Moreover, the mapping ϕ : Pη → R defined by
ϕ(〈α, β, γ〉) = Ψδ
(
Ψδ
(
α
ρ
))
is a one-to-one correspondence.
5. 2 is inert in R. By Theorem 2.1, R =
{
x+ y
√
d
2
∣∣∣∣ x, y ∈ Z and x ≡ y (mod 2)
}
and 2 is
a prime in R. Notice that the norm of 2 in Q(
√
d) is 4, this means that the parity is not as useful as
in the previous sections.
Theorem 5.1. Let η ∈ R and η = uπa11 π
a2
2 . . . πamm , where ak ∈ Z+
0 , u is a unit and πk ∈ R are
non-associate primes such that 2 - πk. Set ρ = πb11 π
b2
2 . . . πbmm , where bk =
⌈ak
2
⌉
. Then Pη is{〈
α,
α2 − η2
2η
,
α2 + η2
2η
〉 ∣∣∣∣ α = τρ for some τ ∈ R, where 2 - τ
}
.
Moreover, the mapping ϕ : Pη → R defined by
ϕ(〈α, β, γ〉) = Ψ2
(
α
ρ
)
is a one-to-one correspondence.
Proof. Suppose 〈α, (α2 − η2)/2η, (α2 + η2)/2η〉 ∈ Pη. Then 2πa11 π
a2
2 . . . πamm |α2 +
+u2π2a11 π2a22 . . . π2amm . Therefore, πa11 π
a2
2 . . . πamm |α2. Hence ρ |α, say α = τρ for some τ ∈ R.
It is easy to see that 2 - τ.
Conversely, suppose α = τρ, where τ ∈ R and 2 - τ. We have η |α2− η2. Let α = (x+ y
√
d)/2
and η = (z + w
√
d)/2, where x, y, z, w ∈ Z and x ≡ y, z ≡ w (mod 2). Since 2 - α and 2 - η,
x ≡ y ≡ z ≡ w ≡ 1 (mod 2). Hence 2 |α−η and thus 2 |α2−η2. Since gcd (η, 2) = 1, 2η |α2+η2.
Theorem 5.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
A RING OF PYTHAGOREAN TRIPLES OVER QUADRATIC FIELDS 139
Theorem 5.2. Let η ∈ R and η = u2a0πa11 π
a2
2 . . . πamm , where a0 ≥ 1 and for k ≥ 1, ak ∈ Z+
0 ,
u is a unit and πk ∈ R are non-associate primes such that 2 - πk. Set ρ = 2b0πb11 π
b2
2 . . . πbmm , where
b0 =
⌈
a0 + 1
2
⌉
and bk =
⌈ak
2
⌉
. Then Pη is
{〈
α,
α2 − η2
2η
,
α2 + η2
2η
〉 ∣∣∣∣ α = τρ for some τ ∈ R
}
.
Moreover, the mapping ϕ : Pη → R defined by
ϕ(〈α, β, γ〉) =
α
ρ
is a one-to-one correspondence.
Proof. The proof is similar to the proof of Theorem 3.2.
6. The ring structure. We combine results from Sections 3 – 5 and define operations addition
and multiplication on Pη and P to establish rings of Pythagorean triples. The ring structures of Pη
and P are constructed from the ring structure of R.
Corollary 6.1. Let η be an algebraic integer. 〈Pη,⊕,�〉 is a commutative ring with identity,
where ⊕ and � are operations on Pη defined by
〈α, β, γ〉 ⊕ 〈µ, ν, λ〉 = ϕ−1(ϕ(〈α, β, γ〉) + ϕ(〈µ, ν, λ〉))
and
〈α, β, γ〉 � 〈µ, ν, λ〉 = ϕ−1(ϕ(〈α, β, γ〉) · ϕ(〈µ, ν, λ〉)).
Corollary 6.2. The mapping Φ: P → R×R given by
Φ(〈α, β, γ〉) = (γ − β, ϕ(〈α, β, γ〉))
is a bijection. Consequently, 〈P,�,�〉 is a commutative ring with identity where � and � are
operations on P defined by
〈α, β, γ〉� 〈µ, ν, λ〉 = Φ−1(Φ(〈α, β, γ〉) + Φ(〈µ, ν, λ〉))
and
〈α, β, γ〉� 〈µ, ν, λ〉 = Φ−1(Φ(〈α, β, γ〉) · Φ(〈µ, ν, λ〉)).
1. Dawson B. A ring of Pythagorean triples // Missouri J. Math. Sci. – 1994. – 6. – P. 72 – 77.
2. Cross J. T. Primitive Pythagorean triples of Gaussian integers // Math. Mag. – 1986. – 59, № 2. – P. 106 – 110.
Received 25.03.12,
after revision — 20.09.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
|
| id | umjimathkievua-article-2117 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:19:03Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cd/b6eaeeee21d89554afa4800f1a5a8bcd.pdf |
| spelling | umjimathkievua-article-21172019-12-05T10:24:29Z A Ring of Pythagorean Triples over Quadratic Fields Кільце пiфагорових трійок над квадратними полями Harnchoowong, A. Somboonkulavudi, C. Гарнчоошонг, А. Сомбоонкулавуді, Ц. Let K be a quadratic field and let R be the ring of integers of K such that R is a unique factorization domain. The set P of all Pythagorean triples in R is partitioned into P η , sets of triples 〈α, β, γ〉 in P where η = γ − β. We show the ring structures of each P η and P from the ring structure of R. Нехай $K$ — квадратне поле, а $R$ — кільцє цілих з $K$ таких, що $R$ — єдина факторизаційна область. Множина $P$ всіх піфагорових трійок з $R$ разбивається на $P_{η}$ , множини трійок $〈α, β, γ〉$ в $P$, де $η = γ − β.$. В роботі показано кільцеві структури для кожного $P_{η}$ та $P$ з кільцевої структури $R$. Institute of Mathematics, NAS of Ukraine 2014-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2117 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 1 (2014); 135–139 Український математичний журнал; Том 66 № 1 (2014); 135–139 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2117/1236 https://umj.imath.kiev.ua/index.php/umj/article/view/2117/1237 Copyright (c) 2014 Harnchoowong A.; Somboonkulavudi C. |
| spellingShingle | Harnchoowong, A. Somboonkulavudi, C. Гарнчоошонг, А. Сомбоонкулавуді, Ц. A Ring of Pythagorean Triples over Quadratic Fields |
| title | A Ring of Pythagorean Triples over Quadratic Fields |
| title_alt | Кільце пiфагорових трійок над квадратними полями |
| title_full | A Ring of Pythagorean Triples over Quadratic Fields |
| title_fullStr | A Ring of Pythagorean Triples over Quadratic Fields |
| title_full_unstemmed | A Ring of Pythagorean Triples over Quadratic Fields |
| title_short | A Ring of Pythagorean Triples over Quadratic Fields |
| title_sort | ring of pythagorean triples over quadratic fields |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2117 |
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