Differences between revisions 27 and 29 (spanning 2 versions)
⇤ ← Revision 27 as of 2012-05-09 12:58:01 →
Size: 30437
Editor: chapoton
Comment: removed j in Computing modular forms (fixed)
← Revision 29 as of 2012-05-09 13:34:14 → ⇥
Size: 30359
Editor: chapoton
Comment: fixed Adding points on an elliptic curve
Deletions are marked like this. Additions are marked like this.
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                    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)],
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    REFERENCES:      REFERENCES:
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        Weisstein, Eric W. "Prime-Generating Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html          Weisstein, Eric W. "Prime-Generating Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
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        elif y<0 and -x >= y and y<x: return 4*(y+1)^2 -11*(y+1) + (start+7) +x          elif y<0 and -x >= y and y<x: return 4*(y+1)^2 -11*(y+1) + (start+7) +x
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    #Takes in an n and the start value of the spiral and gives its (x,y) coordinate      #Takes in an n and the start value of the spiral and gives its (x,y) coordinate
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        num = num - start +1          num = num - start +1
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        top = ceil(sqrt(num))             top = ceil(sqrt(num))
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            else:              else:
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            else:              else:
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    #These functions return an int based on where the t is located in the spiral 
    #These functions return an int based on where the t is located in the spiral
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    if n !=0: x_cord, y_cord = find_xy(n, start) #Overrides the user given x and y coordinates      if n !=0: x_cord, y_cord = find_xy(n, start) #Overrides the user given x and y coordinates
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    if show_lines: 
        for t in [(-size-1)..size+1]: 		     if show_lines:
        for t in [(-size-1)..size+1]:
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            if m.is_pseudoprime(): main_list.add(m)              if m.is_pseudoprime(): main_list.add(m)
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    #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.      #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.
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            else: x-=1              else: x-=1
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        elif county < overcount: 
        elif county < overcount:
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            else: y-=1              else: y-=1
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        else:          else:
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        if not invert and num in main_list: 
        if not invert and num in main_list:
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    if n != 0: 
    if n != 0:
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    else:      else:
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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. 		 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.
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        R = points(list2, alpha = .1) #Faded Composites 
    else: 		         R = points(list2, alpha = .1) #Faded Composites
    else:
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        R=points(list2, hue = .1, pointsize = p_size) 
    		         R=points(list2, hue = .1, pointsize = p_size)
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        Q = plot(W1+W2+W3+W4, alpha = .1)                   Q = plot(W1+W2+W3+W4, alpha = .1)
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        if show_curves:          if show_curves:
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            if n > (floor(sqrt(n)))^2 and n <= (floor(sqrt(n)))^2 + floor(sqrt(n)):              if n > (floor(sqrt(n)))^2 and n <= (floor(sqrt(n)))^2 + floor(sqrt(n)):
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            else:              else:
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            def g(m): return (a*m^2+b*m+c);              def g(m): return (a*m^2+b*m+c);
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            c= c2;              c= c2;
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def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40),  def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40),
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== A Charpoly and Hecke Operator Graph FIXME == == A Charpoly and Hecke Operator Graph ==
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    G = Graph(T, multiedges=True, loops=not three_d)     G = DiGraph(T, multiedges=not three_d)
    if three_d:
        G.remove_loops()
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        MP += line([(i,0),(i,r)], rgbcolor='black')          MP += line([(i,0),(i,r)], rgbcolor='black')
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    S += point(e_pt,pointsize=50, rgbcolor='red')      S += point(e_pt,pointsize=50, rgbcolor='red')
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    S += point(f_gs_pt,pointsize=75, rgbcolor='black')             S += point(f_gs_pt,pointsize=75, rgbcolor='black')
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        S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)),          S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)),
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    S += point(e_pt,pointsize=50, rgbcolor='red')      S += point(e_pt,pointsize=50, rgbcolor='red')
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== Adding points on an elliptic curve FIXME == == Adding points on an elliptic curve ==
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    else:      else:
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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)
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 curve = E.plot(rgbcolor = (0,0,1),xmin=25,xmax=25,plot_points=300)  curve = E.plot(rgbcolor = (0,0,1),xmin=-5,xmax=5,plot_points=300)
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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)
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    """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p,      """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p,

Contents

  1. Integer Factorization
    1. Divisibility Poset
    2. Factor Trees
    3. Factoring an Integer
  2. Prime Numbers
    1. Illustrating the prime number theorem
    2. Prime Spiral - Square FIXME
    3. Prime Spiral - Polar
  3. Modular Forms
    1. Computing modular forms
    2. Computing the cuspidal subgroup
    3. A Charpoly and Hecke Operator Graph
  4. Modular Arithmetic
    1. Quadratic Residue Table FIXME
    2. Cubic Residue Table FIXME
  5. Cyclotomic Fields
    1. Gauss and Jacobi Sums in Complex Plane
    2. Exhaustive Jacobi Plotter
  6. Elliptic Curves
    1. Adding points on an elliptic curve
    2. Plotting an elliptic curve over a finite field
  7. Cryptography
    1. The Diffie-Hellman Key Exchange Protocol
  8. Other
    1. Continued Fraction Plotter FIXME
    2. Computing Generalized Bernoulli Numbers
    3. Fundamental Domains of SL_2(ZZ)

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 FIXME

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

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