Pythagorean Triples: The Hyperbolic View

Beauregard, Raymond A.; Suryanarayan, E. R.

doi:10.1080/07468342.1996.11973772

“…In [5], this result has been generalized from Z to any ring of integers of an algebraic number field. In [3] and [4], the following operation has been studied .…”

Section: Commutative Monoids Of Pythagorean Triplesmentioning

confidence: 99%

“…In [1] and [2], an operation over Pythagorean triples has been studied, showing a group of primitive Pythagorean triples. In [3] and [4], a monoid structure has been provided.…”

Section: Introductionmentioning

confidence: 99%

Groups and monoids of Pythagorean triples connected to conics

Murru

¹

,

Abrate

²

,

Barbero

³

et al. 2017

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: 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.

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“…Since e is a multiple of d, we can write e = kd for a positive integer k. As will be verified in Theorem 1, associating k and h to (a, b, c) sets up a one-to-one correspondence of the PTs with the pairs of positive integers (k, h). For example, everybody's favorite PT (3,4,5) corresponds to the pair (1, 1), and (4,3,5) and (5,12,13) correspond to (1,2) and (2, 1) respectively, while the nonprimitive PTs (48,189,195) and (459,1260,1341) correspond to (7,6) and (21,81). We call this correspondence the height-excess enumeration.…”

Section: Height and Excess Of Pythagorean Triples D A R R Y L M C C Umentioning

confidence: 99%

Height and Excess of Pythagorean Triples

McCullough¹

2005

Mathematics Magazine

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Does the world really need another article about Pythagorean triples? Here is why we think so. The set of Pythagorean triples has a lot of interesting structure, which has intrigued both amateur and professional mathematicians. It is the topic of an extensive mathematical literature, almost all of which relies on an enumeration of primitive Pythagorean triples that has been known since ancient times. But it is not widely known that there is a different enumeration, based on two simple geometric parameters that we call the height and the excess. In this article, we will use these parameters to make some known results about Pythagorean triples more transparent. And we will use them to achieve a better understanding of one natural group structure on the set of primitive Pythagorean triples, and to discover another one.Recall that a Pythagorean triple (PT) is an ordered triple (a, b, c) of positive integers such that a 2 + b 2 = c 2 . When a and b are relatively prime, the triple is a primitive PT (PPT). Each PT is a positive integer multiple of a uniquely determined PPT.The height and excess parameters are shown in FIGURE 1. For a PT (a, b, c), the height h is just c − b, and the excess e is a + b − c. The term excess arises from the fact that e is simply the extra distance one must travel when going along the two legs instead of the hypotenuse. Not all combinations of h and e can occur in an integer-sided triangle. We will see that, for a given h, the possible values of e are exactly the integer multiples of a certain integer d. The integer d is called the increment, and it is related to h in a simple way: d is the smallest positive integer whose square is divisible by 2h. Since e is a multiple of d, we can write e = kd for a positive integer k. As will be verified in Theorem 1, associating k and h to (a, b, c) sets up a one-to-one correspondence of the PTs with the pairs of positive integers (k, h). For example, everybody's favorite PT (3,4,5) corresponds to the pair (1, 1), and (4, 3, 5) and (5, 12, 13) correspond to (1, 2) and (2, 1) respectively, while the nonprimitive PTs (48, 189, 195) and (459, 1260, 1341) correspond to (7, 6) and (21, 81). We call this correspondence the height-excess enumeration.In the rest of this article, we will see various uses of the height and excess parameters. The overarching goal is to find structure on the set of Pythagorean triples. To best understand a particular structure on the set of PTs, we need to view it with the c THE MATHEMATICAL ASSOCIATION OF AMERICA

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“…The height and excess parameters lead to several other systems of coordinates, and we will use whichever of these systems of coordinates seems best for viewing the structure that we are trying to understand. Besides the k-h coordinates coming from the height-excess enumeration, and a closely-related kind of coordinates on PPTs, called k-q coordinates, we will use a-h coordinates, in which (3,4,5) is [3,1], and e-h coordinates, in which (3, 4, 5) is 2, 1 . Each of these coordinate systems reveals some of the structure of the set of PTs that is hidden when the PTs are written in the conventional way.…”

Section: Height and Excess Of Pythagorean Triples D A R R Y L M C C Umentioning

confidence: 99%

“…Since e is a multiple of d, we can write e = kd for a positive integer k. As will be verified in Theorem 1, associating k and h to (a, b, c) sets up a one-to-one correspondence of the PTs with the pairs of positive integers (k, h). For example, everybody's favorite PT (3,4,5) corresponds to the pair (1, 1), and (4, 3, 5) and (5, 12, 13) correspond to (1, 2) and (2, 1) respectively, while the nonprimitive PTs (48, 189, 195) and (459, 1260, 1341) correspond to (7, 6) and (21, 81). We call this correspondence the height-excess enumeration.…”

mentioning

confidence: 99%

Height and Excess of Pythagorean Triples

McCullough

¹

2005

Mathematics Magazine

View full text Add to dashboard Cite

Does the world really need another article about Pythagorean triples? Here is why we think so. The set of Pythagorean triples has a lot of interesting structure, which has intrigued both amateur and professional mathematicians. It is the topic of an extensive mathematical literature, almost all of which relies on an enumeration of primitive Pythagorean triples that has been known since ancient times. But it is not widely known that there is a different enumeration, based on two simple geometric parameters that we call the height and the excess. In this article, we will use these parameters to make some known results about Pythagorean triples more transparent. And we will use them to achieve a better understanding of one natural group structure on the set of primitive Pythagorean triples, and to discover another one.Recall that a Pythagorean triple (PT) is an ordered triple (a, b, c) of positive integers such that a 2 + b 2 = c 2 . When a and b are relatively prime, the triple is a primitive PT (PPT). Each PT is a positive integer multiple of a uniquely determined PPT.The height and excess parameters are shown in FIGURE 1. For a PT (a, b, c), the height h is just c − b, and the excess e is a + b − c. The term excess arises from the fact that e is simply the extra distance one must travel when going along the two legs instead of the hypotenuse. Not all combinations of h and e can occur in an integer-sided triangle. We will see that, for a given h, the possible values of e are exactly the integer multiples of a certain integer d. The integer d is called the increment, and it is related to h in a simple way: d is the smallest positive integer whose square is divisible by 2h. Since e is a multiple of d, we can write e = kd for a positive integer k. As will be verified in Theorem 1, associating k and h to (a, b, c) sets up a one-to-one correspondence of the PTs with the pairs of positive integers (k, h). For example, everybody's favorite PT (3,4,5) corresponds to the pair (1, 1), and (4, 3, 5) and (5, 12, 13) correspond to (1, 2) and (2, 1) respectively, while the nonprimitive PTs (48, 189, 195) and (459, 1260, 1341) correspond to (7, 6) and (21, 81). We call this correspondence the height-excess enumeration.In the rest of this article, we will see various uses of the height and excess parameters. The overarching goal is to find structure on the set of Pythagorean triples. To best understand a particular structure on the set of PTs, we need to view it with the c THE MATHEMATICAL ASSOCIATION OF AMERICA

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Pythagorean Triples: The Hyperbolic View

Cited by 12 publications

References 4 publications

Groups and monoids of Pythagorean triples connected to conics

Groups and monoids of Pythagorean triples connected to conics

Height and Excess of Pythagorean Triples

Height and Excess of Pythagorean Triples

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