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All published worksheets from http://sagenb.org

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Sage Demo and Introduction

Sage is:

  1. A large free open source mathematics software project that I direct.
  2. Web-based at http://sagenb.org or you can run it locally.
  3. Very powerful for number theory, and also combinatorics, graph theory, and numerical analysis (via scipy).
  4. Uses Python for the main user language.

GCD

gcd?

gcd(2010,2007)
3
gcd([15,25,20,45])
5
gcd(90324809238420398423230948203948, 2903480923840923849082309482)
6

How to make up huge random integers:

n = ZZ.random_element(10^10000); m = ZZ.random_element(10^10000)
len(n.digits())
10000
time gcd(n,m)
3 Time: CPU 0.00 s, Wall: 0.00 s
timeit('gcd(n,m)')
125 loops, best of 3: 1.82 ms per loop
2^3
8
%python
2^3
1
%python
1/3
0
preparse('2^3')
'Integer(2)**Integer(3)'
1/3
1/3
preparse('1/3')
'Integer(1)/Integer(3)'
type(1/3)
<type 'sage.rings.rational.Rational'>
parent(1/3)
Rational Field
QQ
Rational Field
i = 0
for a in QQ:
    print a
    i += 1
    if i > 100: break
0 1 -1 1/2 -1/2 2 -2 1/3 -1/3 3 -3 2/3 -2/3 3/2 -3/2 1/4 -1/4 4 -4 3/4 -3/4 4/3 -4/3 1/5 -1/5 5 -5 2/5 -2/5 5/2 -5/2 3/5 -3/5 5/3 -5/3 4/5 -4/5 5/4 -5/4 1/6 -1/6 6 -6 5/6 -5/6 6/5 -6/5 1/7 -1/7 7 -7 2/7 -2/7 7/2 -7/2 3/7 -3/7 7/3 -7/3 4/7 -4/7 7/4 -7/4 5/7 -5/7 7/5 -7/5 6/7 -6/7 7/6 -7/6 1/8 -1/8 8 -8 3/8 -3/8 8/3 -8/3 5/8 -5/8 8/5 -8/5 7/8 -7/8 8/7 -8/7 1/9 -1/9 9 -9 2/9 -2/9 9/2 -9/2 4/9 -4/9 9/4 -9/4 5/9 -5/9

Enumerating Primes

prime_range(47)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43]
prime_range(200)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199]
prime_range(100,200)
[101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199]
@interact
def f(n=(10..2500), clr=Color('red')):
    print prime_range(n)

Sieving Primes

@interact
def f(n=(2..400), pmax=(2..20)):
    P = [2]
    X = [3,5,..,n]
    while True:
        if len(X) == 0: break
        p = X[0]
        if p > pmax:
            P.extend(X); break
        P.append(p)
        X = [a for a in X if a%p]
    print "<html>"
    for a in [1..n]:
        clr = '#a00' if a in P else '#aaf'
        print "<font color='%s'>%s</font>"%(clr, ' '*(3-len(str(a)))+str(a)),
        if a%20 == 0:
            print
    print "</html>"

Mersenne Primes

The prime below is the largest known prime.  The GIMPS project found it, winning them a \$100,000 prize from the EFF, since it has > ten million digits.

time p = 2^43112609 - 1
Time: CPU 0.01 s, Wall: 0.01 s
time v = p.digits()
Time: CPU 14.26 s, Wall: 14.44 s
len(v)
12978189

Last 10 digits:

v[-10:]
[3, 9, 6, 2, 0, 7, 4, 6, 1, 3]

First 10 digits:

v[:10]
[1, 1, 5, 2, 5, 1, 7, 9, 6, 6]

Frequency histogram of first nn digits:

@interact
def f(n=(10,100,..10^5)):
    time T = finance.TimeSeries(v[:n])
    show(T.plot_histogram())

Counting Primes

prime_pi
Function that counts the number of primes up to x
prime_pi(10^9)
50847534
for n in [10,100,1000,10000,100000]:
    print n, prime_pi(n)
10 4 100 25 1000 168 10000 1229 100000 9592
@interact
def f(n=500):
    show(plot(prime_pi, 1, n) + plot(x/(log(x)-1), (x,10,n), color='red') )

@interact
def f(n=500):
    show(plot(lambda x: prime_pi(x) - x/(log(x)-1), 10, n))

Wait, that looks like a nice clean smooth curve?  What is it "basically" a plot of?   

The Riemann Hypothesis is a conjectural answer to this question...