scholarly journals Groups and monoids of Pythagorean triples connected to conics

2017 ◽  
Vol 15 (1) ◽  
pp. 1323-1331
Author(s):  
Nadir Murru ◽  
Marco Abrate ◽  
Stefano Barbero ◽  
Umberto Cerruti
Keyword(s):  
Commutative Monoid ◽  
Commutative Monoids ◽  
Pythagorean Triples ◽  
Pythagorean Triple

Abstract We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and 3 × 3 matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations.

THE SHUFFLE VARIANT OF JEŚMANOWICZ’ CONJECTURE CONCERNING PYTHAGOREAN TRIPLES

2011 ◽  
Vol 90 (3) ◽  
pp. 355-370
Author(s):  
TAKAFUMI MIYAZAKI
Keyword(s):  
Unique Solution ◽  
Pythagorean Triples ◽  
Pythagorean Triple ◽  
Positive Integers

AbstractLet (a,b,c) be a primitive Pythagorean triple such that b is even. In 1956, Jeśmanowicz conjectured that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in the positive integers. This is one of the most famous unsolved problems on Pythagorean triples. In this paper we propose a similar problem (which we call the shuffle variant of Jeśmanowicz’ problem). Our problem states that the equation cx+by=az with x,y and z positive integers has the unique solution (x,y,z)=(1,1,2) if c=b+1 and has no solutions if c>b+1 . We prove that the shuffle variant of the Jeśmanowicz problem is true if c≡1 mod b.

SOME NOTES ON A GENERALIZED VERSION OF PYTHAGOREAN TRIPLES

2020 ◽  
Vol 4 (2) ◽  
pp. 103
Author(s):  
Leomarich F Casinillo ◽  
Emily L Casinillo
Keyword(s):  
Diophantine Equation ◽  
Pythagorean Triples ◽  
Pythagorean Triple ◽  
Positive Integers

A Pythagorean triple is a set of three positive integers a, b and c that satisfy the Diophantine equation a^2+b^2=c^2. The triple is said to be primitive if gcd(a, b, c)=1 and each pair of integers and  are relatively prime, otherwise known as non-primitive. In this paper, the generalized version of the formula that generates primitive and non-primitive Pythagorean triples that depends on two positive integers  k and n, that is, P_T=(a(k, n), b(k, n), c(k, n)) were constructed. Further, we determined the values of  k and n that generates primitive Pythagorean triples and give some important results.

PRESENTATIONS OF FINITELY GENERATED SUBMONOIDS OF FINITELY GENERATED COMMUTATIVE MONOIDS

2002 ◽  
Vol 12 (05) ◽  
pp. 659-670 ◽  
Author(s):  
J. C. ROSALES ◽  
P. A. GARCÍA-SÁNCHEZ ◽  
J. I. GARCÍA-GARCÍA
Keyword(s):  
Commutative Monoid ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Algorithmic Method ◽  
Simple Equations ◽  
Torsion Free

We give an algorithmic method for computing a presentation of any finitely generated submonoid of a finitely generated commutative monoid. We use this method also for calculating the intersection of two congruences on ℕp and for deciding whether or not a given finitely generated commutative monoid is t-torsion free and/or separative. The last section is devoted to the resolution of some simple equations on a finitely generated commutative monoid.

scholarly journals A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples

2021 ◽  
Vol 5 (1) ◽  
pp. 115-127
Author(s):  
Van Thien Nguyen ◽  
◽  
Viet Kh. Nguyen ◽  
Pham Hung Quy ◽  
◽  
...  
Keyword(s):  
Positive Integer ◽  
Simple Proof ◽  
Diophantine Equation ◽  
Necessary Conditions ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Pythagorean Triples ◽  
Pythagorean Triple

Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2, b=2uv, c=u^2+v^2\), where \(u>v>0\) are co-prime and not of the same parity. In 1956, L. Jesmanowicz conjectured that for any positive integer \(n\), the Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only the positive integer solution \((x,y,z)=(2,2,2)\). In this connection we call a positive integer solution \((x,y,z)\ne (2,2,2)\) with \(n>1\) exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a unified simple proof of the theorem of Le-Yang-Fu. Next we give necessary conditions for the existence of exceptional solutions in the case \(v=2,\ u\) is an odd prime. As an application we show the truth of the Jesmanowicz conjecture for all prime values \(u < 100\).

scholarly journals Irreducible decomposition of binomial ideals

2016 ◽  
Vol 152 (6) ◽  
pp. 1319-1332 ◽  
Author(s):  
Thomas Kahle ◽  
Ezra Miller ◽  
Christopher O’Neill
Keyword(s):  
Number Theory ◽  
Duke Math ◽  
Commutative Monoid ◽  
Commutative Monoids ◽  
Irreducible Decomposition ◽  
Binomial Ideal ◽  
Binomial Ideals ◽  
Mesoprimary Decomposition

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

scholarly journals Pythagorean Triples with Common Sides

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Raymond Calvin Ochieng ◽  
Chiteng’a John Chikunji ◽  
Vitalis Onyango-Otieno
Keyword(s):  
Finite Number ◽  
Pythagorean Triples ◽  
Pythagorean Triple

There exist a finite number of Pythagorean triples that have a common leg. In this paper we derive the formulas that generate pairs of primitive Pythagorean triples with common legs and also show the process of how to determine all the primitive and nonprimitive Pythagorean triples for a given leg of a Pythagorean triple.

scholarly journals ON PRESENTATIONS OF COMMUTATIVE MONOIDS

1999 ◽  
Vol 09 (05) ◽  
pp. 539-553 ◽  
Author(s):  
J. C. ROSALES ◽  
P. A. GARCÍA-SÁNCHEZ ◽  
J. M. URBANO-BLANCO
Keyword(s):  
Commutative Monoid ◽  
Commutative Monoids ◽  
Minimal Presentations

In this paper, we introduce the concept of a strongly reduced monoid and we characterize the minimal presentations for such monoids. As a consequence, we give a method to obtain a presentation for any commutative monoid.

A Matrix Method for Generating Pythagorean Triples

1987 ◽  
Vol 80 (2) ◽  
pp. 103-108
Author(s):  
Phyllis Lefton
Keyword(s):  
Matrix Method ◽  
Greatest Common Divisor ◽  
Pythagorean Triples ◽  
Pythagorean Triple

This article describes a program that uses an interesting matrix method to generate Pythagorean triples -that is, solutions of the equation a2 + b2 = c2 for which a, b, and c are integers. Only primitive triples are found, that is, those for which a > 0, b > 0, c > 0, and the greatest common divisor of a, b, and c is one. This result suffices because nonprimitive triples are just multiples of primitive ones. We shall use the abbreviation PPT for primitive Pythagorean triple.

A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING PRIMITIVE PYTHAGOREAN TRIPLES

2016 ◽  
Vol 95 (1) ◽  
pp. 5-13 ◽  
Author(s):  
MOU-JIE DENG ◽  
DONG-MING HUANG
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Character Theory ◽  
Positive Integer Solution ◽  
Pythagorean Triples ◽  
Pythagorean Triple ◽  
Positive Integers ◽  
Baker's Theorem

Let $a,b,c$ be a primitive Pythagorean triple and set $a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$, where $m$ and $n$ are positive integers with $m>n$, $\text{gcd}(m,n)=1$ and $m\not \equiv n~(\text{mod}~2)$. In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ is $(x,y,z)=(2,2,2)$. We use biquadratic character theory to investigate the case with $(m,n)\equiv (2,3)~(\text{mod}~4)$. We show that Jeśmanowicz’ conjecture is true in this case if $m+n\not \equiv 1~(\text{mod}~16)$ or $y>1$. Finally, using these results together with Laurent’s refinement of Baker’s theorem, we show that Jeśmanowicz’ conjecture is true if $(m,n)\equiv (2,3)~(\text{mod}~4)$ and $n<100$.

scholarly journals The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid

2019 ◽  
Vol 19 (07) ◽  
pp. 2050137 ◽  
Author(s):  
Felix Gotti
Keyword(s):  
Finite Rank ◽  
Convex Cone ◽  
The Other ◽  
Extremal System ◽  
Commutative Monoid ◽  
Commutative Monoids ◽  
Other Hand ◽  
Sets Of Lengths

Let [Formula: see text] be an atomic monoid. For [Formula: see text], let [Formula: see text] denote the set of all possible lengths of factorizations of [Formula: see text] into irreducibles. The system of sets of lengths of [Formula: see text] is the set [Formula: see text]. On the other hand, the elasticity of [Formula: see text], denoted by [Formula: see text], is the quotient [Formula: see text] and the elasticity of [Formula: see text] is the supremum of the set [Formula: see text]. The system of sets of lengths and the elasticity of [Formula: see text] both measure how far [Formula: see text] is from being half-factorial, i.e. [Formula: see text] for each [Formula: see text]. Let [Formula: see text] denote the collection comprising all submonoids of finite-rank free commutative monoids, and let [Formula: see text]. In this paper, we study the system of sets of lengths and the elasticity of monoids in [Formula: see text]. First, we construct for each [Formula: see text] a monoid in [Formula: see text] having extremal system of sets of lengths. It has been proved before that the system of sets of lengths does not characterize (up to isomorphism) monoids in [Formula: see text]. Here we use our construction to extend this result to [Formula: see text] for any [Formula: see text]. On the other hand, it has been recently conjectured that the elasticity of any monoid in [Formula: see text] is either rational or infinite. We conclude this paper by proving that this is indeed the case for monoids in [Formula: see text] and for any monoid in [Formula: see text] whose corresponding convex cone is polyhedral.

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