Definition. A Pythagorean triple (x, y, z) is a triple of positive integers where x² + y² = 2². This can be thought of as describing an x x y rectangle with the property that the diagonal z is also of integer length. A Pythagorean triple (x, y, z) is primitive if x, y, z are coprime (i.e. there is no integer k> 1 which divides all of them). (a) Write a Python function PrimPyth (n) which returns a list of primitive Pythagorean triples (x, y, z) where 0

Definition. A Pythagorean triple (x, y, z) is a triple of positive integers where x² + y² = 2². This can be thought of as describing an x x y rectangle with the property that the diagonal z is also of integer length. A Pythagorean triple (x, y, z) is primitive if x, y, z are coprime (i.e. there is no integer k> 1 which divides all of them). (a) Write a Python function PrimPyth (n) which returns a list of primitive Pythagorean triples (x, y, z) where 0

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Topological Sort

Topological sort is a sorting technique that sets up a hierarchy in a process. In this technique, when a directed network has nodes connected by arrows, the root node is placed before the successor node.

Question

Definition. A Pythagorean triple (x, y, z) is a triple of positive integers where x² + y² = z². This can be
thought of as describing an x × y rectangle with the property that the diagonal z is also of integer length.
A Pythagorean triple (x, y, z) is primitive if x, y, z are coprime (i.e. there is no integer k > 1 which divides all
of them).
(a) Write a Python function PrimPyth (n) which returns a list of primitive Pythagorean triples (x, y, z) where
0 < x < y < z <n. For example, PrimPyth (6) should return [(3,4,5)]
Hint: In this project we represent triples as tuples (x, y, z) not as lists [x,y,z]. The two data-types
behave similarly in many ways. Do not write a triply nested loop which searches through all triples (x, y, z)
as this will take n³ steps! Instead, use the fact (proved by Euclid) that every primitive Pythagorean triple
arises as (m² — n², 2mn, m² + n²) where m, n are coprime integers which are not both odd. (Of course the
first two entries may be swapped.)
(b) Run your function from part (a) with n = 10000 and plot a scattergraph of y against x.
(c) Write a function Pyth (n) which returns a list of (not necessarily primitive) Pythagorean triples (x, y, z)
where z <n.
Hint: Use your function from part (a).
(d) Run your function from part (c) with n= 10000 and plot a scattergraph of y against x. (This should be a
new figure not overwriting the one from part (b)).
(e) Comment on your the features of your two graphs, writing your answer as a comment.

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