Lab1template.sagews

#Lab 1: Plotting legs (a,b) of Primitive Pythagorean Triples
reset

<function reset at 0x7f3837eaf050>

#1. The function makes a list of pairs [m,n] of rationals m/n from N diagonals of the table of rationals
#//Input:?
#//Output:?
def mn_pairs(N):
    L=[];
    for n in range(2,N+2):
        temp=[[m,n] for m in range(1,n)]
        L.extend(temp)
    return L

#Example
L=mn_pairs(10);L

[[1, 2], [1, 3], [2, 3], [1, 4], [2, 4], [3, 4], [1, 5], [2, 5], [3, 5], [4, 5], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6], [1, 7], [2, 7], [3, 7], [4, 7], [5, 7], [6, 7], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9], [1, 10], [2, 10], [3, 10], [4, 10], [5, 10], [6, 10], [7, 10], [8, 10], [9, 10], [1, 11], [2, 11], [3, 11], [4, 11], [5, 11], [6, 11], [7, 11], [8, 11], [9, 11], [10, 11]]

# 2.The function makes a list of legs [a,b] of PTs and excludes PTs that are not primitive (i.e., pairs with a and b are not mutuallly prime)
#//Input:
#//Output:number of PTs in L,number of PPT in L,list of PPTs
def ppt_legs(L):
    p_triples=[[c[1]^2-c[0]^2,2*c[0]*c[1]] for c in L ]#list of PTs
    pp_triples=[t for t in p_triples if gcd(t[0],t[1]) == 1]#list of PPTs
    return(len(p_triples),len(pp_triples),pp_triples)

#Example continued

n_pt,n_ppt,S=ppt_legs(L);n_pt #Convenient way to get multiple output

55

n_ppt

27

#Example of plotting points in SageMath
point(S,size=30,color='red',figsize=[4,3])

#3. Main code
#//Input: Number of diagonals in the table of rationals
#//Output:?
def plot_ppt(N):
    L=mn_pairs(N)
    n_pt,n_ppt,S=ppt_legs(L)
    G=point(S,size=30,color='red',figsize=[5,4])
    G.show()
    return("Number of PTs:", n_pt, "Number of PTs:",n_ppt)

plot_ppt(20)

('Number of PTs:', 210, 'Number of PTs:', 92)

#Example of alternative plotting points in S using Python libraries
# Get data for plotting points
x=[s[0] for s in S];y=[s[1] for s in S]

import matplotlib.pyplot as plt
from matplotlib.pyplot import figure
figure(num=None, figsize=(5, 3))
figure=plt.plot(x, y,'ro',markersize=4)
plt.show(figure)

#Advanced nice snippet of code for plotting (just FYI)
import numpy as np
import matplotlib.pyplot as plt

#evenly sampled time at 200ms intervals
t = np.arange(0., 5., 0.2)

# red dashes, blue squares and green triangles
plt.plot(t, t, 'r--', t, t**2, 'bs', t, t**3, 'g^')
plt.show()

[<matplotlib.lines.Line2D object at 0x7fd118429510>, <matplotlib.lines.Line2D object at 0x7fd118429650>, <matplotlib.lines.Line2D object at 0x7fd118429dd0>]

#For instructor_________________________________
#Data for detecting missed points
xblue=[195,187,171,115,75,27];yblue=[28,84,140,252,308,364]
#figure=plt.plot(xblue,yblue,'bo',markersize=4)# Include into the plotting command code above

#4 Find the missing point in the figure created in Part 3 (see directions in Chapter 2)
gcd(147,196)

49

#5 Write a brief summary (3-4 sentences). See directions in Chapter 2.
︠db102b61-bfa2-48cd-8ebf-7beeb0fd6662︠