Sage Demo

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.
{{{id=11| /// }}}

GCD

{{{id=13| gcd(2010,2007) /// 3 }}} {{{id=14| gcd([15,25,20,45]) /// 5 }}} {{{id=10| gcd(90324809238420398423230948203948, 2903480923840923849082309482) /// 6 }}}

How to make up huge random integers:

{{{id=16| n = ZZ.random_element(10^10000); m = ZZ.random_element(10^10000) len(n.digits()) /// 10000 }}} {{{id=15| gcd(n,m) /// 1 }}} {{{id=18| /// }}}

Enumerating Primes

{{{id=6| prime_range(50) /// [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47] }}} {{{id=5| 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] }}} {{{id=4| @interact def f(n=(10..2500)): print prime_range(n) /// }}} {{{id=21| /// }}} {{{id=3| /// }}}

Sieving Primes

{{{id=1| @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 "" for a in [1..n]: clr = '#a00' if a in P else '#aaf' print "%s"%(clr, ' '*(3-len(str(a)))+str(a)), if a%20 == 0: print print "" /// }}} {{{id=24| /// }}}

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.

{{{id=26| time p = 2^43112609 - 1 /// Time: CPU 0.01 s, Wall: 0.01 s }}} {{{id=27| time v = p.digits() /// Time: CPU 14.40 s, Wall: 14.49 s }}} {{{id=28| len(v) /// 12978189 }}}

Last 10 digits:

{{{id=22| v[-10:] /// [3, 9, 6, 2, 0, 7, 4, 6, 1, 3] }}}

First 10 digits:

{{{id=32| v[:10] /// [1, 1, 5, 2, 5, 1, 7, 9, 6, 6] }}}

Frequency histogram of first $n$ digits:

{{{id=36| @interact def f(n=(10,100,..10^5)): time T = finance.TimeSeries(v[:n]) show(T.plot_histogram()) ///
n 
}}}

Counting Primes

{{{id=33| prime_pi /// Function that counts the number of primes up to x }}} {{{id=40| prime_pi(10) /// 4 }}} {{{id=41| for n in [10,100,1000,10000,100000]: print n, prime_pi(n) /// 10 4 100 25 1000 168 10000 1229 100000 9592 }}} {{{id=42| @interact def f(n=(10..5000)): show(plot(prime_pi, 1, n)) ///
n 
}}} {{{id=45| /// }}}

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...

{{{id=44| /// }}} {{{id=43| /// }}}