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= Sage Interactions - Number Theory =
goto [:interact:interact main page]
<<TableOfContents>>

= Integer Factorization =

== Divisibility Poset ==
by William Stein
{{{#!sagecell
@interact
def _(n=(5..100)):
    Poset(([1..n], lambda x, y: y%x == 0) ).show()
}}}

{{attachment:divposet.png}}
Line 6: Line 18:
{{{ {{{#!sagecell
Line 39: Line 51:
                    g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)],                      g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)],
Line 51: Line 63:
attachment:factortree.png

=== Continued Fraction Plotter ===
{{attachment:factortree.png}}

More complicated demonstration using Mathematica: http://demonstrations.wolfram.com/FactorTrees/

== Factoring an Integer ==
by Timothy Clemans

Sage implementation of the Mathematica demonstration of the same name. http://demonstrations.wolfram.com/FactoringAnInteger/

{{{#!sagecell
@interact
def _(r=selector(range(0,10000,1000), label='range', buttons=True), n=slider(0,1000,1,2,'n',False)):
    if not r and n in (0, 1):
        n = 2
    s = '$%d = %s$' % (r + n, factor(r + n))
    s = s.replace('*', '\\times')
    html(s)
}}}

= Prime Numbers =

== Illustrating the prime number theorem ==
Line 55: Line 86:
{{{
@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
{{{
{{{#!sagecell
Line 71: Line 92:
attachment:primes.png

=== Computing Generalized Bernoulli Numbers ===
{{attachment:primes.png}}

== Prime Spiral - Square FIXME ==
by David Runde
{{{#!sagecell
@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 [email protected]
        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 = copy(M)
        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
{{{#!sagecell
@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}}


= Modular Forms =

== Computing modular forms ==
by William Stein
{{{#!sagecell
@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 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
{{{#!sagecell
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

{{{#!sagecell
# 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 = DiGraph(T, multiedges=not three_d)
    if three_d:
        G.remove_loops()
    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}}

= Modular Arithmetic =

== Quadratic Residue Table FIXME ==
by Emily Kirkman
{{{#!sagecell
from numpy import array as narray
@interact
def quad_res_plot(first_n_odd_primes = (20,200),display_size=[7..15]):

    # Compute numpy matrix of legendre symbols
    r = int(first_n_odd_primes)
    np = [nth_prime(i+2) for i in range(r)]
    leg = [[legendre_symbol(np[i], np[j]) for i in range(r)] for j in range(r)]
    na = narray(leg)
    for i in range(r):
        for j in range(r):
            if na[i][j] == 1 and Mod((np[i]-1)*(np[j]-1)//4,2) == 0:
                na[i][j] = 2
    m = matrix(na)

    # Define plot structure
    MP = matrix_plot(m, cmap='Oranges')
    for i in range(r):
        if np[-1] < 100:
            MP += text('%d'%nth_prime(i+2),(-.75,r-i-.5), rgbcolor='black')
            MP += text('%d'%nth_prime(i+2), (i+.5,r+.5), rgbcolor='black')
        if len(np) < 75:
            MP += line([(0,i),(r,i)], rgbcolor='black')
            MP += line([(i,0),(i,r)], rgbcolor='black')
    if np[-1] < 100:
        for i in range(r): # rows
            for j in range(r): # cols
                if m[j][i] == 0:
                    MP += text('0',(i+.5,r-j-.5),rgbcolor='black')
                elif m[j][i] == -1:
                    MP += text('N',(i+.5,r-j-.5),rgbcolor='black')
                elif m[j][i] == 1:
                    MP += text('A',(i+.5,r-j-.5),rgbcolor='black')
                elif m[j][i] == 2:
                    MP += text('S',(i+.5,r-j-.5),rgbcolor='black')
    MP += line([(0,r),(r,r)], rgbcolor='black')
    MP += line([(r,0),(r,r)], rgbcolor='black')
    MP += line([(0,0),(r,0)], rgbcolor='black')
    MP += line([(0,0),(0,r)], rgbcolor='black')
    if len(np) < 75:
        MP += text('q',(r/2,r+2), rgbcolor='black', fontsize=15)
        MP += text('p',(-2.5,r/2), rgbcolor='black', fontsize=15)
    MP.show(axes=False, ymax=r, figsize=[display_size,display_size])
    html('Symmetry of Prime Quadratic Residues mod the first %d odd primes.'%r)
}}}

{{attachment:quadres.png}}

{{attachment:quadresbig.png}}

== Cubic Residue Table FIXME ==
by Emily Kirkman
{{{#!sagecell
def power_residue_symbol(alpha, p, m):
    if p.divides(alpha): return 0
    if not p.is_prime():
        return prod(power_residue_symbol(alpha,ell,m)^e
                for ell, e in p.factor())
    F = p.residue_field()
    N = p.norm()
    r = F(alpha)^((N-1)/m)
    k = p.number_field()
    for kr in k.roots_of_unity():
        if r == F(kr):
            return kr


def cubic_is_primary(n):
    g = n.gens_reduced()[0]
    a,b = g.polynomial().coefficients()
    return Mod(a,3)!=0 and Mod(b,3)==0


from numpy import array as narray
@interact
def cubic_sym(n=(10..35),display_size=[7..15]):

    # Compute numpy matrix of primary cubic residue symbols
    r = n
    m=3
    D.<w> = NumberField(x^2+x+1)
    it = D.primes_of_degree_one_iter()
    pp = []
    while len(pp) < r:
        k = it.next()
        if cubic_is_primary(k):
            pp.append(k)
    n = narray([ [ power_residue_symbol(pp[i].gens_reduced()[0], pp[j], m) \
                        for i in range(r) ] for j in range(r) ])

    # Convert to integer matrix for gradient colors
    for i in range(r):
        for j in range(r):
            if n[i][j] == w:
                n[i][j] = int(-1)
            elif n[i][j] == w^2:
                n[i][j] = int(-2)
            elif n[i][j] == 1:
                n[i][j] = int(1)
    m = matrix(n)

    # Define plot structure
    MP = matrix_plot(m,cmap="Blues")
    for i in range(r):
        MP += line([(0,i),(r,i)], rgbcolor='black')
        MP += line([(i,0),(i,r)], rgbcolor='black')
        for j in range(r):
            if m[i][j] == -2:
                MP += text('$\omega^2$',(i+.5,r-j-.5),rgbcolor='black')
            if m[i][j] == -1:
                MP += text('$\omega $',(i+.5,r-j-.5),rgbcolor='black')
            if m[i][j] == 0:
                MP += text('0',(i+.5,r-j-.5),rgbcolor='black')
            if m[i][j] == 1:
                MP += text('R',(i+.5,r-j-.5),rgbcolor='white')
    MP += line([(0,r),(r,r)], rgbcolor='black')
    MP += line([(r,0),(r,r)], rgbcolor='black')
    MP += line([(0,0),(r,0)], rgbcolor='black')
    MP += line([(0,0),(0,r)], rgbcolor='black')
    MP += text('$ \pi_1$',(r/2,r+2), rgbcolor='black', fontsize=25)
    MP += text('$ \pi_2$',(-2.5,r/2), rgbcolor='black', fontsize=25)

    html('Symmetry of Primary Cubic Residues mod ' \
          + '%d primary primes in $ \mathbf Z[\omega]$.'%r)
    MP.show(axes=False, figsize=[display_size,display_size])
}}}

{{attachment:cubres.png}}

= Cyclotomic Fields =

== Gauss and Jacobi Sums in Complex Plane ==
by Emily Kirkman
{{{#!sagecell
def jacobi_sum(e,f):
    # If they are both trivial, return p
    if e.is_trivial() and f.is_trivial():
        return (e.parent()).order() + 1

    # If they are inverses of each other, return -e(-1)
    g = e*f
    if g.is_trivial():
        return -e(-1)

    # If both are nontrivial, apply mult. formula:
    elif not e.is_trivial() and not f.is_trivial():
        return e.gauss_sum()*f.gauss_sum()/g.gauss_sum()

    # If exactly one is trivial, return 0
    else:
        return 0


def latex2(e):
    return latex(list(e.values_on_gens()))


def jacobi_plot(p, e_index, f_index, with_text=True):
    # Set values
    G = DirichletGroup(p)
    e = G[e_index]
    f = G[f_index]
    ef = e*f
    js = jacobi_sum(e,f)
    e_gs = e.gauss_sum()
    f_gs = f.gauss_sum()
    ef_gs = (e*f).gauss_sum()

    # Compute complex coordinates
    f_pt = list(f.values_on_gens()[0].complex_embedding())
    e_pt = list(e.values_on_gens()[0].complex_embedding())
    ef_pt = list(ef.values_on_gens()[0].complex_embedding())
    f_gs_pt = list(f_gs.complex_embedding())
    e_gs_pt = list(e_gs.complex_embedding())
    ef_gs_pt = list(ef_gs.complex_embedding())
    try:
        js = int(js)
        js_pt = [CC(js)]
    except:
        js_pt = list(js.complex_embedding())

    # Define plot structure
    S = circle((0,0),1,rgbcolor='yellow')
    S += line([e_pt,e_gs_pt], rgbcolor='red', thickness=4)
    S += line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3)
    S += line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2)
    S += point(e_pt,pointsize=50, rgbcolor='red')
    S += point(f_pt,pointsize=50, rgbcolor='blue')
    S += point(ef_pt,pointsize=50,rgbcolor='purple')
    S += point(f_gs_pt,pointsize=75, rgbcolor='black')
    S += point(e_gs_pt,pointsize=75, rgbcolor='black')
    S += point(ef_gs_pt,pointsize=75, rgbcolor='black')
    S += point(js_pt,pointsize=100,rgbcolor='green')
    if with_text:
        S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)),
            (3,2.5),fontsize=15, rgbcolor='black')
    else:
        html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js)))

    return S

@interact
def single_jacobi_plot(p=prime_range(3,100), e_range=(0..100), f_range=(0..100)):
    e_index = floor((p-2)*e_range/100)
    f_index = floor((p-2)*f_range/100)
    S = jacobi_plot(p,e_index,f_index,with_text=False)
    S.show(aspect_ratio=1)
}}}

{{attachment:jacobising.png}}

== Exhaustive Jacobi Plotter ==
by Emily Kirkman
{{{#!sagecell
def jacobi_sum(e,f):
    # If they are both trivial, return p
    if e.is_trivial() and f.is_trivial():
        return (e.parent()).order() + 1

    # If they are inverses of each other, return -e(-1)
    g = e*f
    if g.is_trivial():
        return -e(-1)

    # If both are nontrivial, apply mult. formula:
    elif not e.is_trivial() and not f.is_trivial():
        return e.gauss_sum()*f.gauss_sum()/g.gauss_sum()

    # If exactly one is trivial, return 0
    else:
        return 0


def latex2(e):
    return latex(list(e.values_on_gens()))


def jacobi_plot(p, e_index, f_index, with_text=True):
    # Set values
    G = DirichletGroup(p)
    e = G[e_index]
    f = G[f_index]
    ef = e*f
    js = jacobi_sum(e,f)
    e_gs = e.gauss_sum()
    f_gs = f.gauss_sum()
    ef_gs = (e*f).gauss_sum()

    # Compute complex coordinates
    f_pt = list(f.values_on_gens()[0].complex_embedding())
    e_pt = list(e.values_on_gens()[0].complex_embedding())
    ef_pt = list(ef.values_on_gens()[0].complex_embedding())
    f_gs_pt = list(f_gs.complex_embedding())
    e_gs_pt = list(e_gs.complex_embedding())
    ef_gs_pt = list(ef_gs.complex_embedding())
    try:
        js = int(js)
        js_pt = [CC(js)]
    except:
        js_pt = list(js.complex_embedding())

    # Define plot structure
    S = circle((0,0),1,rgbcolor='yellow')
    S += line([e_pt,e_gs_pt], rgbcolor='red', thickness=4)
    S += line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3)
    S += line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2)
    S += point(e_pt,pointsize=50, rgbcolor='red')
    S += point(f_pt,pointsize=50, rgbcolor='blue')
    S += point(ef_pt,pointsize=50,rgbcolor='purple')
    S += point(f_gs_pt,pointsize=75, rgbcolor='black')
    S += point(e_gs_pt,pointsize=75, rgbcolor='black')
    S += point(ef_gs_pt,pointsize=75, rgbcolor='black')
    S += point(js_pt,pointsize=100,rgbcolor='green')
    if with_text:
        S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)),
            (3,2.5),fontsize=15, rgbcolor='black')
    else:
        html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js)))

    return S

@interact
def exhaustive_jacobi_plot(p=prime_range(3,8)):
    ga = [jacobi_plot(p,i,j) for i in range(p-1) for j in range(p-1)[i:]]

    for i in range(len(ga)):
        ga[i].save('j%d.png'%i,figsize=4,aspect_ratio=1,
                    xmin=-2.5,xmax=5, ymin=-2.5, ymax=2.5)

    # Since p is odd, will have n = p-1 even. So 1+2+...+n = (n/2)*(n+1).
    # We divide this by rows of 2.
    rows = ceil(p*(p-1)/4)
    s='<table bgcolor=lightgrey cellpadding=2>'
    for i in range(rows):
        s+='<tr><td align="center"><img src="cell://j%d.png"></td>'%(2*i)
        s+='<td align="center"><img src="cell://j%d.png"></td></tr>'%(2*i+1)
    s+='</table>'
    html(s)}}}

{{attachment:jacobiexh.png}}

= Elliptic Curves =

== Adding points on an elliptic curve ==
by David Møller Hansen
{{{#!sagecell
def point_txt(P,name,rgbcolor):
    if (P.xy()[1]) < 0:
        r = text(name,[float(P.xy()[0]),float(P.xy()[1])-1],rgbcolor=rgbcolor)
    elif P.xy()[1] == 0:
        r = text(name,[float(P.xy()[0]),float(P.xy()[1])+1],rgbcolor=rgbcolor)
    else:
        r = text(name,[float(P.xy()[0]),float(P.xy()[1])+1],rgbcolor=rgbcolor)
    return r

E = EllipticCurve('37a')
list_of_points = E.integral_points()
html("Graphical addition of two points $P$ and $Q$ on the curve $ E: %s $"%latex(E))

def line_from_curve_points(E,P,Q,style='-',rgb=(1,0,0),length=25):
 """
 P,Q two points on an elliptic curve.
 Output is a graphic representation of the straight line intersecting with P,Q.
 """
 # The function tangent to P=Q on E
 if P == Q:
  if P[2]==0:
   return line([(1,-length),(1,length)],linestyle=style,rgbcolor=rgb)
  else:
   # Compute slope of the curve E in P
   l=-(3*P[0]^2 + 2*E.a2()*P[0] + E.a4() - E.a1()*P[1])/((-2)*P[1] - E.a1()*P[0] - E.a3())
   f(x) = l * (x - P[0]) + P[1]
   return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb)
 # Trivial case of P != R where P=O or R=O then we get the vertical line from the other point
 elif P[2] == 0:
  return line([(Q[0],-length),(Q[0],length)],linestyle=style,rgbcolor=rgb)
 elif Q[2] == 0:
  return line([(P[0],-length),(P[0],length)],linestyle=style,rgbcolor=rgb)
 # Non trivial case where P != R
 else:
  # Case where x_1 = x_2 return vertical line evaluated in Q
  if P[0] == Q[0]:
   return line([(P[0],-length),(P[0],length)],linestyle=style,rgbcolor=rgb)

  #Case where x_1 != x_2 return line trough P,R evaluated in Q"
  l=(Q[1]-P[1])/(Q[0]-P[0])
  f(x) = l * (x - P[0]) + P[1]
  return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb)

@interact
def _(P=selector(list_of_points,label='Point P'),Q=selector(list_of_points,label='Point Q'), marked_points = checkbox(default=True,label = 'Points'), Lines = selector([0..2],nrows=1), Axes=True):
 curve = E.plot(rgbcolor = (0,0,1),xmin=-5,xmax=5,plot_points=300)
 R = P + Q
 Rneg = -R
 l1 = line_from_curve_points(E,P,Q)
 l2 = line_from_curve_points(E,R,Rneg,style='--')
 p1 = plot(P,rgbcolor=(1,0,0),pointsize=40)
 p2 = plot(Q,rgbcolor=(1,0,0),pointsize=40)
 p3 = plot(R,rgbcolor=(1,0,0),pointsize=40)
 p4 = plot(Rneg,rgbcolor=(1,0,0),pointsize=40)
 textp1 = point_txt(P,"$P$",rgbcolor=(0,0,0))
 textp2 = point_txt(Q,"$Q$",rgbcolor=(0,0,0))
 textp3 = point_txt(R,"$P+Q$",rgbcolor=(0,0,0))
 if Lines==0:
  g=curve
 elif Lines ==1:
  g=curve+l1
 elif Lines == 2:
  g=curve+l1+l2
 if marked_points:
  g=g+p1+p2+p3+p4
 if P != Q:
  g=g+textp1+textp2+textp3
 else:
  g=g+textp1+textp3
 g.axes_range(xmin=-5,xmax=5,ymin=-13,ymax=13)
 show(g,axes = Axes)
}}}
{{attachment:PointAddEllipticCurve.png}}


== Plotting an elliptic curve over a finite field ==
{{{#!sagecell
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}}

= Cryptography =

== The Diffie-Hellman Key Exchange Protocol ==
by Timothy Clemans and William Stein
{{{#!sagecell
@interact
def diffie_hellman(bits=slider(8, 513, 4, 8, 'Number of bits', False),
    button=selector(["Show new example"],label='',buttons=True)):
    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)

    html("""
<style>
.gamodp, .gbmodp {
color:#000;
padding:5px
}
.gamodp {
background:#846FD8
}
.gbmodp {
background:#FFFC73
}
.dhsame {
color:#000;
font-weight:bold
}
</style>
<h2 style="color:#000;font-family:Arial, Helvetica, sans-serif">%s-Bit Diffie-Hellman Key Exchange</h2>
<ol style="color:#000;font-family: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>
    """ % (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}}

= Other =

== Continued Fraction Plotter ==
by William Stein
{{{#!sagecell
@interact
def _(number=e, ymax=selector([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}}

== Computing Generalized Bernoulli Numbers ==
Line 75: Line 824:
{{{ {{{#!sagecell
Line 91: Line 840:
attachment:bernoulli.png


=== Fundamental Domains of SL_2(ZZ) ===
{{attachment:bernoulli.png}}


== Fundamental Domains of SL_2(ZZ) ==
Line 96: Line 845:
{{{ {{{#!sagecell
Line 104: Line 853:
def _(gen = selector(['t+1', 't-1', '-1/t'], nrows=1)): def _(gen = selector(['t+1', 't-1', '-1/t'], buttons=True,nrows=1)):
Line 121: Line 870:
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
{{attachment:fund_domain.png}}

= Multiple Zeta Values =
by Akhilesh P.
== Word to composition ==
{{{#!sagecell
@interact
def _( weight=(7,(2..30))):
 n=weight
 a=[0 for i in range(n-1)]
 a.append(1)
 @interact
 def _(v=('word', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x))))):
  a=[v[i] for i in range(len(v))]
  def bintocomp(a):
 b=[]
 count=1
 for j in range(len(a)):
  if(a[j]==0):
   count=count+1
  else:
   b.append(count)
   count=1
 return(b)
  print "Composition is ",bintocomp(a)
}}}

{{attachment:akhi2.png}}
== Composition to Word ==
{{{#!sagecell
@interact
def _( Depth=(7,(1..30))):
 n=Depth
 a=[]
 a.append(2)
 a=a+[1 for i in range(1,n)]
 @interact
 def _(v=('composition', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x))))):
  a=[v[i] for i in range(len(v))]
  def comptobin(a):
 word=[]
 for i in range(len(a)):
  word=word+[0]*(a[i]-1)+[1]
 return(word)

  print "Word is ",comptobin(a)
}}}

{{attachment:akhi3.png}}


== Computing Multiple Zeta values ==
=== Word Input ===
{{{#!sagecell
R=RealField(10)
@interact
def _( weight=(5,(2..20))):
 n=weight
 a=[0 for i in range(n-1)]
 a.append(1)
 @interact
 def _(v=('word', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x)))), accuracy=(100..100000)):
  D=accuracy
  a=[v[i] for i in range(len(v))]
  DD=int(3.321928*D)+int(R(log(3.321928*D))/R(log(10)))+4
  RIF=RealIntervalField(DD)
  def Li(word):
        n=int(DD*log(10)/log(2))+1
        B=[]
        L=[]
        S=[]
        count=-1
        k=len(word)
        for i in range(k):
                B.append(RIF('0'))
                L.append(RIF('0'))
                if(word[i]==1 and i<k-1):
                        S.append(RIF('0'))
                        count=count+1
        T=RIF('1')
        for m in range(n):
                T=T/2
                B[k-1]=RIF('1')/(m+1)
                j=count
                for i in range(k-2,-1,-1):
                        if(word[i]==0):
                                B[i]=B[i+1]/(m+1)
                        elif(word[i]==1):
                                B[i]=S[j]/(m+1)
                                S[j]=S[j]+B[i+1]
                                j=j-1
                        L[i]=T*B[i]+L[i]
                L[k-1]=T*B[k-1]+L[k-1]
        return(L)
  def dual(a):
        b=list()
        b=a
        b=b[::-1]
        for i in range(len(b)):
                b[i]=1-b[i]
        return(b)
  def zeta(a):
        b=dual(a)
        l1=Li(a)+[1]
        l2=Li(b)+[1]
        Z=RIF('0')
        for i in range(len(l1)):
                Z=Z+l1[i]*l2[len(a)-i]
        return(Z)
  print zeta(a)
}}}
{{attachment:akhi1.png}}
=== Composition Input ===
{{{#!sagecell
R=RealField(10)
@interact
def _( Depth=(5,(2..20))):
 n=Depth
 a=[2]
 a=a+[1 for i in range(n-1)]
 @interact
 def _(v=('Composition', input_grid(1, n, default=[a], to_value=lambda x: vector(flatten(x)))), accuracy=(100..100000)):
  D=accuracy
  a=[v[i] for i in range(len(v))]
  def comptobin(a):
        word=[]
        for i in range(len(a)):
                word=word+[0]*(a[i]-1)+[1]
        return(word)
  a=comptobin(a)
  DD=int(3.321928*D)+int(R(log(3.321928*D))/R(log(10)))+4
  RIF=RealIntervalField(DD)
  def Li(word):
        n=int(DD*log(10)/log(2))+1
        B=[]
        L=[]
        S=[]
        count=-1
        k=len(word)
        for i in range(k):
                B.append(RIF('0'))
                L.append(RIF('0'))
                if(word[i]==1 and i<k-1):
                        S.append(RIF('0'))
                        count=count+1
        T=RIF('1')
        for m in range(n):
                T=T/2
                B[k-1]=RIF('1')/(m+1)
                j=count
                for i in range(k-2,-1,-1):
                        if(word[i]==0):
                                B[i]=B[i+1]/(m+1)
                        elif(word[i]==1):
                                B[i]=S[j]/(m+1)
                                S[j]=S[j]+B[i+1]
                                j=j-1
                        L[i]=T*B[i]+L[i]
                L[k-1]=T*B[k-1]+L[k-1]
        return(L)
  def dual(a):
        b=list()
        b=a
        b=b[::-1]
        for i in range(len(b)):
                b[i]=1-b[i]
        return(b)
  def zeta(a):
        b=dual(a)
        l1=Li(a)+[1]
        l2=Li(b)+[1]
        Z=RIF('0')
        for i in range(len(l1)):
                Z=Z+l1[i]*l2[len(a)-i]
        return(Z)

  print zeta(a)
}}}
{{attachment:akhi5.png}}

Integer Factorization

Divisibility Poset

by William Stein

divposet.png

Factor Trees

by William Stein

factortree.png

More complicated demonstration using Mathematica: http://demonstrations.wolfram.com/FactorTrees/

Factoring an Integer

by Timothy Clemans

Sage implementation of the Mathematica demonstration of the same name. http://demonstrations.wolfram.com/FactoringAnInteger/

Prime Numbers

Illustrating the prime number theorem

by William Stein

primes.png

Prime Spiral - Square FIXME

by David Runde

SquareSpiral.PNG

Prime Spiral - Polar

by David Runde

PolarSpiral.PNG

Modular Forms

Computing modular forms

by William Stein

modformbasis.png

Computing the cuspidal subgroup

by William Stein

cuspgroup.png

A Charpoly and Hecke Operator Graph

by William Stein

heckegraph.png

Modular Arithmetic

Quadratic Residue Table FIXME

by Emily Kirkman

quadres.png

quadresbig.png

Cubic Residue Table FIXME

by Emily Kirkman

cubres.png

Cyclotomic Fields

Gauss and Jacobi Sums in Complex Plane

by Emily Kirkman

jacobising.png

Exhaustive Jacobi Plotter

by Emily Kirkman

jacobiexh.png

Elliptic Curves

Adding points on an elliptic curve

by David Møller Hansen

PointAddEllipticCurve.png

Plotting an elliptic curve over a finite field

ellffplot.png

Cryptography

The Diffie-Hellman Key Exchange Protocol

by Timothy Clemans and William Stein

dh.png

Other

Continued Fraction Plotter

by William Stein

contfracplot.png

Computing Generalized Bernoulli Numbers

by William Stein (Sage-2.10.3)

bernoulli.png

Fundamental Domains of SL_2(ZZ)

by Robert Miller

fund_domain.png

Multiple Zeta Values

by Akhilesh P.

Word to composition

akhi2.png

Composition to Word

akhi3.png

Computing Multiple Zeta values

Word Input

akhi1.png

Composition Input

akhi5.png

interact/number_theory (last edited 2020-06-14 09:10:48 by chapoton)