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Editor: jason
Comment:
Deletions are marked like this. Additions are marked like this.
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== Divisibility Poset ==
by William Stein
{{{#!sagecell
@interact
def _(n=(5..100)):
    Poset(([1..n], lambda x, y: y%x == 0) ).show()
}}}

{{attachment:divposet.png}}
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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== Prime Spiral - Square == == Prime Spiral - Square FIXME ==
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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== Computing modular forms == == Computing modular forms FIXME ==
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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== A Charpoly and Hecke Operator Graph == == A Charpoly and Hecke Operator Graph FIXME ==
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{{{ {{{#!sagecell
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== Quadratic Residue Table == == Quadratic Residue Table FIXME ==
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{{{ {{{#!sagecell
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== Cubic Residue Table == == Cubic Residue Table FIXME ==
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{{{ {{{#!sagecell
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    if Mod(a,3)!=0 and Mod(b,3)==0:
        return True
    else:
        return False		     return Mod(a,3)!=0 and Mod(b,3)==0
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{{{ {{{#!sagecell
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    S = circle((0,0),1,rgbcolor='yellow')  \
    + line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) \
    + line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) \
    + line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) \
    + point(e_pt,pointsize=50, rgbcolor='red')  \
    + point(f_pt,pointsize=50, rgbcolor='blue') \
    + point(ef_pt,pointsize=50,rgbcolor='purple') \
    + point(f_gs_pt,pointsize=75, rgbcolor='black') \
    + point(e_gs_pt,pointsize=75, rgbcolor='black') \
    + point(ef_gs_pt,pointsize=75, rgbcolor='black') \
    + point(js_pt,pointsize=100,rgbcolor='green')		     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')
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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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{{{ {{{#!sagecell
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    S = circle((0,0),1,rgbcolor='yellow')  \
    + line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) \
    + line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) \
    + line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) \
    + point(e_pt,pointsize=50, rgbcolor='red')  \
    + point(f_pt,pointsize=50, rgbcolor='blue') \
    + point(ef_pt,pointsize=50,rgbcolor='purple') \
    + point(f_gs_pt,pointsize=75, rgbcolor='black') \
    + point(e_gs_pt,pointsize=75, rgbcolor='black') \
    + point(ef_gs_pt,pointsize=75, rgbcolor='black') \
    + point(js_pt,pointsize=100,rgbcolor='green')		     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')
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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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        ga[i].save('j%d.PNG'%i,figsize=4,aspect_ratio=1, \         ga[i].save('j%d.png'%i,figsize=4,aspect_ratio=1,
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    html('<table bgcolor=lightgrey cellpadding=2>')     s='<table bgcolor=lightgrey cellpadding=2>'
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        html('<tr><td align="center"><img src="cell://j%d.PNG"></td>'%(2*i))
        html('<td align="center"><img src="cell://j%d.PNG"></td></tr>'%(2*i+1))
    html('</table>')
}}}		         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)}}}
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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    print """
<html>		     html("""
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</ol></html> </ol>
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       (g^ b)^a, g^a, b, p, (g^a)^b)        (g^ b)^a, g^a, b, p, (g^a)^b))
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell
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{{{ {{{#!sagecell

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 FIXME
    2. Computing the cuspidal subgroup
    3. A Charpoly and Hecke Operator Graph FIXME
  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
    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 FIXME

by William Stein

modformbasis.png

Computing the cuspidal subgroup

by William Stein

cuspgroup.png

A Charpoly and Hecke Operator Graph FIXME

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

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