Sage Interactions - Number Theory

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TableOfContents

Factor Trees

by William Stein

import random
def ftree(rows, v, i, F):
    if len(v) > 0: # add a row to g at the ith level.
        rows.append(v)
    w = []
    for i in range(len(v)):
        k, _, _ = v[i]
        if k is None or is_prime(k):
            w.append((None,None,None))
        else:
            d = random.choice(divisors(k)[1:-1])
            w.append((d,k,i))
            e = k//d
            if e == 1:
                w.append((None,None))
            else:
                w.append((e,k,i))
    if len(w) > len(v):
        ftree(rows, w, i+1, F)
def draw_ftree(rows,font):
    g = Graphics()
    for i in range(len(rows)):
        cur = rows[i]
        for j in range(len(cur)):
            e, f, k = cur[j]
            if not e is None:
                if is_prime(e):
                     c = (1,0,0)
                else:
                     c = (0,0,.4)
                g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c)
                if not k is None and not f is None:
                    g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)], 
                    alpha=0.5)
    return g

@interact
def factor_tree(n=100, font=(10, (8..20)), redraw=['Redraw']):
    n = Integer(n)
    rows = []
    v = [(n,None,0)]
    ftree(rows, v, 0, factor(n))
    show(draw_ftree(rows, font), axes=False)

attachment:factortree.png

Continued Fraction Plotter

by William Stein

@interact
def _(number=e, ymax=selector([None,5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]):
    c = list(continued_fraction(RealField(prec)(number))); print c
    show(line([(i,z) for i, z in enumerate(c)],rgbcolor=clr),ymax=ymax,figsize=[10,2])

attachment:contfracplot.png

Illustrating the prime number thoerem

by William Stein

@interact
def _(N=(100,(2..2000))):
    html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N)
    show(plot(prime_pi, 0, N, rgbcolor='red') + plot(x/(log(x)-1), 5, N, rgbcolor='blue'))

attachment:primes.png

Computing Generalized Bernoulli Numbers

by William Stein (Sage-2.10.3)

@interact
def _(m=selector([1..15],nrows=2), n=(7,(3..10))):
    G = DirichletGroup(m)
    s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,m)
    s += '<table border=1>'
    s += '<tr bgcolor="#edcc9c"><td align=center>$\\chi$</td><td>Conductor</td>' + \
           ''.join('<td>$B_{%s,\chi}$</td>'%k for k in [1..n]) + '</tr>'
    for eps in G.list():
        v = ''.join(['<td align=center bgcolor="#efe5cd">$%s$</td>'%latex(eps.bernoulli(k)) for k in [1..n]])
        s += '<tr><td bgcolor="#edcc9c">%s</td><td bgcolor="#efe5cd" align=center>%s</td>%s</tr>\n'%(
             eps, eps.conductor(), v)
    s += '</table>'
    html(s)

attachment:bernoulli.png

Fundamental Domains of SL_2(ZZ)

by Robert Miller

L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000, -1, -1)]
R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000)]
xes = [x/1000.0 for x in xrange(-500,501,1)]
M = [[x,abs(sqrt(x^2-1))] for x in xes]
fundamental_domain = L+M+R
fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
@interact
def _(gen = selector(['t+1', 't-1', '-1/t'], nrows=1)):
    global fundamental_domain
    if gen == 't+1':
        fundamental_domain = [[x+1,y] for x,y in fundamental_domain]
    elif gen == 't-1':
        fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
    elif gen == '-1/t':
        new_dom = []
        for x,y in fundamental_domain:
            sq_mod = x^2 + y^2
            new_dom.append([(-1)*x/sq_mod, y/sq_mod])
        fundamental_domain = new_dom
    P = polygon(fundamental_domain)
    P.ymax(1.2); P.ymin(-0.1)
    P.show()

attachment:fund_domain.png

Computing modular forms

by William Stein

j = 0
@interact
def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40), 
      group=[(Gamma0, 'Gamma0'), (Gamma1, 'Gamma1')]):
    M = CuspForms(group(N),k)
    print j; global j; j += 1
    print M; print '\n'*3
    print "Computing basis...\n\n"
    if M.dimension() == 0:
         print "Space has dimension 0"
    else:
        prec = max(prec, M.dimension()+1)
        for f in M.basis():
             view(f.q_expansion(prec))
    print "\n\n\nDone computing basis."

attachment:modformbasis.png

Computing the cuspidal subgroup

by William Stein

html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>')
@interact
def _(N=selector([1..8*13], ncols=8, width=10, default=10)):
    A = J0(N)
    print A.cuspidal_subgroup()

attachment:cuspgroup.png

A Charpoly and Hecke Operator Graph

by William Stein

# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots
@interact
def f(N = prime_range(11,400),
      p = selector(prime_range(2,12),nrows=1),
      three_d = ("Three Dimensional", False)):
    S = SupersingularModule(N)
    T = S.hecke_matrix(p)
    G = Graph(T, multiedges=True, loops=not three_d)
    html("<h1>Charpoly and Hecke Graph: Level %s, T_%s</h1>"%(N,p))
    show(T.charpoly().factor())
    if three_d:
        show(G.plot3d(), aspect_ratio=[1,1,1])
    else:
        show(G.plot(),figsize=7)

attachment:heckegraph.png

Demonstrating the Diffie-Hellman Key Exchange Protocol

by Timothy Clemans (refereed by William Stein)

@interact
def diffie_hellman(button=selector(["New example"],label='',buttons=True), 
    bits=("Number of bits of prime", (8,12,..512))):
    maxp = 2^bits
    p = random_prime(maxp)
    k = GF(p)
    if bits>100:
        g = k(2)
    else:
        g = k.multiplicative_generator()
    a = ZZ.random_element(10, maxp)
    b = ZZ.random_element(10, maxp)

    print """
<html>
<style>
.gamodp {
background:yellow
}
.gbmodp {
background:orange
}
.dhsame {
color:green;
font-weight:bold
}
</style>
<h2>%s-Bit Diffie-Hellman Key Exchange</h2>
<ol style="color:#000;font:12px Arial, Helvetica, sans-serif">
<li>Alice and Bob agree to use the prime number p=%s and base g=%s.</li>
<li>Alice chooses the secret integer a=%s, then sends Bob (<span class="gamodp">g<sup>a</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gamodp">%s</span>.</li>
<li>Bob chooses the secret integer b=%s, then sends Alice (<span class="gbmodp">g<sup>b</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gbmodp">%s</span>.</li>
<li>Alice computes (<span class="gbmodp">g<sup>b</sup> mod p</span>)<sup>a</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
<li>Bob computes (<span class="gamodp">g<sup>a</sup> mod p</span>)<sup>b</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
</ol></html>
    """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p, 
       (g^ b)^a, g^a, b, p, (g^a)^b)

attachment:dh.png

Plotting an elliptic curve over a finite field

E = EllipticCurve('37a')
@interact
def _(p=slider(prime_range(1000), default=389)):
    show(E)
    print "p = %s"%p
    show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0)

attachment:ellffplot.png

Prime Spiral - Square

by David Runde

@interact
def square_prime_spiral(start=1, end=100, size_limit = 10, show_lines=false, invert=false, x_cord=0, y_cord=0, n = 0):

    """
    REFERENCES: 
        Alpern, Dario. "Ulam's Spiral". http://www.alpertron.com.ar/ULAM.HTM
        Sacks, Robert. http://www.NumberSpiral.com
        Ventrella, Jeffery. "Prime Numbers are the Holes Behind Complex Composite Patterns". http://www.divisorplot.com
        Williamson, John. Number Spirals. http://www.dcs.gla.ac.uk/~jhw/spirals/index.html jhw@dcs.gla.ac.uk
        Weisstein, Eric W. "Prime-Generating Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html 
    """

    #Takes an (x,y) coordinate (and the start of the spiral) and gives its corresponding n value
    def find_n(x,y, start):
        if x>0 and y>-x and y<=x: return 4*(x-1)^2 + 5*(x-1) + (start+1) + y
        elif x<=0 and y>=x and y<=-x: return 4*x^2 - x + (start) -y
        elif y>=0 and -y+1 <= x and y-1 >= x: return 4*y^2 -y + start -x
        elif y<0 and -x >= y and y<x: return 4*(y+1)^2 -11*(y+1) + (start+7) +x 
        else: print 'NaN'

    #Takes in an n and the start value of the spiral and gives its (x,y) coordinate 
    def find_xy(num, start):
        num = num - start +1 
        bottom = floor(sqrt(num))
        top = ceil(sqrt(num))    
        if bottom^2 < num and num<=bottom^2+bottom+1:
            if bottom%2 == 0:
                x=-bottom/2
                y=-x-(num-bottom^2)+1
            else: 
                x=bottom/2+1/2
                y=-x + (num-bottom^2)
        else:
            if top%2 == 0:
                y=top/2
                x=-top/2+1+top^2-num
            else: 
                y=-top/2+1/2
                x=top/2 -1/2 - (top^2-num)
        x = Integer(x)
        y = Integer(y)
        return (x,y)

    if start < 1 or end <=start: print "invalid start or end value"
    if n > end: print "WARNING: n is larger than the end value"

    #Changes the entry of a matrix by taking the old matrix and the (x,y) coordinate (in matrix coordinates) and returns the changed matrix
    def matrix_morph(M, x, y, set):
        N = M.copy()
        N[x-1,y] = set
        M = N
        return M
 
    #These functions return an int based on where the t is located in the spiral 
    def SW_NE(t, x, y, start):
        if -y<x: return 4*t^2 + 2*t -x+y+start
        else: return 4*t^2 + 2*t +x-y+start
    def NW_SE(t, x, y, start):
        if x<y: return 4*t^2 -x-y+start
        else: return 4*t^2 + 4*t +x+y+start

    size = ceil(sqrt(end-start+1)) #Size of the matrix
    num=copy(start) # Start number (might not be used)
    x = ceil(size/2)   #starting center x of the matrix (in matrix coordinates)
    y = copy(x)        #starting center y of the matrix (in matrix coordinates)
    if n !=0: x_cord, y_cord = find_xy(n, start) #Overrides the user given x and y coordinates 
    xt = copy(x_cord)
    yt = copy(y_cord)
    countx=0
    county=0
    overcount = 1
    if size <= size_limit: M = matrix(ZZ, size+1) # Allows the numbers to be seen in the smaller matricies
    else: M = matrix(GF(2), size+1) # Restricts the entries to 0 or 1
    
    main_list = set()
    #print x_cord, y_cord
    if show_lines: 
        for t in [(-size-1)..size+1]: 
            m= SW_NE(t, xt, yt, start)
            if m.is_pseudoprime(): main_list.add(m) 
            m= NW_SE(t, xt, yt, start)
            if m.is_pseudoprime(): main_list.add(m)
    else: main_list = set(prime_range(end))

    #This for loop changes the matrix by spiraling out from the center and changing each entry as it goes. It is faster than the find_xy function above. 
    for num in [start..end]:
        #print x, "=x  y=", y, " num =", num
        if countx < overcount:
            if overcount % 2 == 1: x+=1
            else: x-=1 
            countx += 1
        
        elif county < overcount: 
            if overcount % 2 == 1: y+=1
            else: y-=1 
            county += 1
        else: 
            overcount += 1
            countx=2
            county=0
            if overcount % 2 == 1: x+=1
            else: x-=1
    
        if not invert and num in main_list: 
            if size <= size_limit: M = matrix_morph(M, x, y, num)
            else: M = matrix_morph(M, x, y, 1)

        elif invert and num not in main_list: #This does the opposite of the above if statement by changing the matrix only when a number is not in the list of allowable primes
            if size <= size_limit: M = matrix_morph(M, x, y, num)
            else: M = matrix_morph(M, x, y, 1)
    
    if n != 0: 
        print '(to go from x,y coords to an n, reset by setting n=0)'
        (x_cord, y_cord) = find_xy(n, start)
        #print 'if n =', n, 'then (x,y) =', (x_cord, y_cord)

    print '(x,y) =', (x_cord, y_cord), '<=> n =', find_n(x_cord, y_cord, start)
    print ' '
    print "SW/NE line"
    if -y_cord<x_cord: print '4*t^2 + 2*t +', -x_cord+y_cord+start
    else: print '4*t^2 + 2*t +', +x_cord-y_cord+start

    print "NW/SE line"
    if x_cord<y_cord: print '4*t^2 +', -x_cord-y_cord+start
    else: print '4*t^2 + 4*t +', +x_cord+y_cord+start

    if size <= size_limit: show(M) #Displays the matrix with integer entries
    else: 
        M.visualize_structure() # Displays the final resulting matrix as a series of pixels (1 <=> pixel on)
        #matrix_plot(M)

attachment:SquareSpiral.PNG

Prime Spiral - Polar

by David Runde

@interact
def polar_prime_spiral(start=1, end=2000, show_factors = false, highlight_primes = false, show_curves=true, n = 0): 

    #For more information about the factors in the spiral, visit http://www.dcs.gla.ac.uk/~jhw/spirals/index.html by John Williamson. 

    if start < 1 or end <=start: print "invalid start or end value"
    if n > end: print "WARNING: n is greater than end value"
    def f(n):
        return (sqrt(n)*cos(2*pi*sqrt(n)), sqrt(n)*sin(2*pi*sqrt(n)))
    
    list =[]
    list2=[]
    if show_factors == false:
        for i in [start..end]:
            if i.is_pseudoprime(): list.append(f(i-start+1)) #Primes list
            else: list2.append(f(i-start+1)) #Composites list
        P = points(list)
        R = points(list2, alpha = .1) #Faded Composites 
    else: 
        for i in [start..end]:
            list.append(disk((f(i-start+1)),0.05*pow(2,len(factor(i))-1), (0,2*pi))) #resizes each of the dots depending of the number of factors of each number
            if i.is_pseudoprime() and highlight_primes: list2.append(f(i-start+1))
        P = plot(list)
        p_size = 5 #the orange dot size of the prime markers
        if not highlight_primes: list2 = [(f(n-start+1))]
        R=points(list2, hue = .1, pointsize = p_size) 
    
    if n > 0:
        print 'n =', factor(n)
        
        p = 1
    #The X which marks the given n
        W1 = disk((f(n-start+1)), p, (pi/6, 2*pi/6))
        W2 = disk((f(n-start+1)), p, (4*pi/6, 5*pi/6))
        W3 = disk((f(n-start+1)), p, (7*pi/6, 8*pi/6))
        W4 = disk((f(n-start+1)), p, (10*pi/6, 11*pi/6))
        Q = plot(W1+W2+W3+W4, alpha = .1)          

        n=n-start+1        #offsets the n for different start values to ensure accurate plotting
        if show_curves: 
            begin_curve = 0
            t = var('t')
            a=1
            b=0
            if n > (floor(sqrt(n)))^2 and n <= (floor(sqrt(n)))^2 + floor(sqrt(n)): 
                c = -((floor(sqrt(n)))^2 - n)
                c2= -((floor(sqrt(n)))^2 + floor(sqrt(n)) - n)
            else: 
                c = -((ceil(sqrt(n)))^2 - n)
                c2= -((floor(sqrt(n)))^2 + floor(sqrt(n)) - n)
            print 'Pink Curve:  n^2 +', c
            print 'Green Curve: n^2 + n +', c2
            def g(m): return (a*m^2+b*m+c); 
            def r(m) : return sqrt(g(m))
            def theta(m) : return r(m)- m*sqrt(a)
            S1 = parametric_plot(((r(t))*cos(2*pi*(theta(t))),(r(t))*sin(2*pi*(theta(t)))), begin_curve, ceil(sqrt(end-start)), rgbcolor=hue(0.8), thickness = .2) #Pink Line

            b=1
            c= c2; 
            S2 = parametric_plot(((r(t))*cos(2*pi*(theta(t))),(r(t))*sin(2*pi*(theta(t)))), begin_curve, ceil(sqrt(end-start)), rgbcolor=hue(0.6), thickness = .2) #Green Line

            show(R+P+S1+S2+Q, aspect_ratio = 1, axes = false)
        else: show(R+P+Q, aspect_ratio = 1, axes = false)
    else: show(R+P, aspect_ratio = 1, axes = false)

attachment:PolarSpiral.PNG