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1 3 2п + 1, y = 2n? + 2n, z = 2n2 + 2n +1, (Pythagoras) = 2n, y = n? – 1, z = n² + 1, (Plato) 2mп, y = m² – n², z = m? + n², (Euclid)

1 3 2п + 1, y = 2n? + 2n, z = 2n2 + 2n +1, (Pythagoras) = 2n, y = n? – 1, z = n² + 1, (Plato) 2mп, y = m² – n², z = m? + n², (Euclid)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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prove for any natural numbers m, n (x,y,z) where ...(equations given in image) are Pythagorean triples. For each of these formulas, give 3 Pythagorean triples that can be obtained from them. Thank you. 

The image shows three sets of formulas attributed to Pythagoras, Plato, and Euclid, which describe integer solutions (x, y, z) for generating Pythagorean triples. These are triplets of positive integers that can be the lengths of the sides of a right-angled triangle, satisfying the equation \(x^2 + y^2 = z^2\). 1. **Pythagoras**: - \( x = 2n + 1 \) - \( y = 2n^2 + 2n \) - \( z = 2n^2 + 2n + 1 \) 2. **Plato**: - \( x = 2n \) - \( y = n^2 - 1 \) - \( z = n^2 + 1 \) 3. **Euclid**: - \( x = 2mn \) - \( y = m^2 - n^2 \) - \( z = m^2 + n^2 \) These formulas are used to derive sets of integers that form Pythagorean triples. Each set provides a method to generate such triples, using parameters \(n\) and \(m\), which are typically positive integers.
Transcribed Image Text:The image shows three sets of formulas attributed to Pythagoras, Plato, and Euclid, which describe integer solutions (x, y, z) for generating Pythagorean triples. These are triplets of positive integers that can be the lengths of the sides of a right-angled triangle, satisfying the equation \(x^2 + y^2 = z^2\). 1. **Pythagoras**: - \( x = 2n + 1 \) - \( y = 2n^2 + 2n \) - \( z = 2n^2 + 2n + 1 \) 2. **Plato**: - \( x = 2n \) - \( y = n^2 - 1 \) - \( z = n^2 + 1 \) 3. **Euclid**: - \( x = 2mn \) - \( y = m^2 - n^2 \) - \( z = m^2 + n^2 \) These formulas are used to derive sets of integers that form Pythagorean triples. Each set provides a method to generate such triples, using parameters \(n\) and \(m\), which are typically positive integers.
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