Proof Without Words: Parametric Representation of Primitive Pythagorean Triples

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

doi:10.1080/0025570x.1996.11996425

Cited by 3 publications

(21 citation statements)

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“…The integers u and v, or simply (u, v) in the formula (9), are called the generating numbers or generators of the triple (a, b, c). Neugebauer claims that this sufficiency part was known to the Babylonians.…”

Section: Characterizationmentioning

confidence: 99%

“…The converse, i.e., showing that any Pythagorean triple is necessarily of the form (9), is more difficult. The earliest record for some special cases of the proof of the converse can be found in the solutions of Problems 8 and 9 of Book II of the Arithmetica of Diophantus.…”

Section: Characterizationmentioning

confidence: 99%

“…The following proof (see Beauregard and Suryanarayan [9] and Joyce [10]) of the converse of Euclid's statement is short and elementary.…”

Section: Characterizationmentioning

confidence: 99%

“…The Pythagorean triple (13500, 12709, 18541) in Row 4, Table 2, can be obtained by letting u = 125, v = 54 in (9). It is interesting to note that the corresponding u and v for each row in Table 2 have no prime factors other than 2, 3, and 5 (the prime divisors of the Babylonian scale 60), and v < 60.…”

Section: Characterizationmentioning

confidence: 99%

“…For u = n + 1, v = n, n ≥ 1, Formula (9) reduces to (4). Indeed, then we have (9) reduces to (7). Indeed, then we have 2uv = 2n, u…”

Section: Characterizationmentioning

confidence: 99%

See 4 more Smart Citations

Pythagorean Triples before and after Pythagoras

Agarwal

2020

Computation

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Following the corrected chronology of ancient Hindu scientists/mathematicians, in this article, a sincere effort is made to report the origin of Pythagorean triples. We shall account for the development of these triples from the period of their origin and list some known astonishing directions. Although for researchers in this field, there is not much that is new in this article, we genuinely hope students and teachers of mathematics will enjoy this article and search for new directions/patterns.

show abstract

Section: Characterizationmentioning

confidence: 99%

Section: Characterizationmentioning

confidence: 99%

“…The following proof (see Beauregard and Suryanarayan [9] and Joyce [10]) of the converse of Euclid's statement is short and elementary.…”

Section: Characterizationmentioning

confidence: 99%

Section: Characterizationmentioning

confidence: 99%

“…For u = n + 1, v = n, n ≥ 1, Formula (9) reduces to (4). Indeed, then we have (9) reduces to (7). Indeed, then we have 2uv = 2n, u…”

Section: Characterizationmentioning

confidence: 99%

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Pythagorean Triples before and after Pythagoras

Agarwal

2020

Computation

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A Novel Approach for Studying Pythagorean Triples Suitable for Students at all Educational Levels

Amato

2024

Eur. J. Pure Appl. Math.

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This paper seeks to showcase how a new approach can breathe new life into research within the traditional domain of Pythagorean triples, introducing innovative applications to invigorate the field. This serves not only as an exemplar but also as a wellspring of inspiration for students at both school and university levels. The demonstrations will underscore that, with fundamental mathematical concepts and unencumbered by intricate calculations, one can unveil novel results and applications with ease. The new results and applications, along with those found in the Preliminary Results section, show how the field of Pythagorean triples is still interesting and stimulating to study, despite the centuries that have elapsed.

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Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

Agarwal¹

2021

JAMP

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In this article we shall examine several different types of figurative numbers which have been studied extensively over the period of 2500 years, and currently scattered on hundreds of websites. We shall discuss their computation through simple recurrence relations, patterns and properties, and mutual relationships which have led to curious results in the field of elementary number theory. Further, for each type of figurative numbers we shall show that the addition of first finite numbers and infinite addition of their inverses often require new/strange techniques. We sincerely hope that besides experts, students and teachers of mathematics will also be benefited with this article.

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Proof Without Words: Parametric Representation of Primitive Pythagorean Triples

Cited by 3 publications

References 0 publications

Pythagorean Triples before and after Pythagoras

Pythagorean Triples before and after Pythagoras

A Novel Approach for Studying Pythagorean Triples Suitable for Students at all Educational Levels

Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

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