Differences between revisions 55 and 66 (spanning 11 versions)
⇤ ← Revision 55 as of 2014-12-23 02:05:59 →
Size: 45948
Editor: akhi
Comment:
← Revision 66 as of 2020-06-04 19:10:25 → ⇥
Size: 48486
Editor: kcrisman
Comment:
Deletions are marked like this. Additions are marked like this.
Line 79: Line 79:
    html(s)     pretty_print(html(s))
Line 89: Line 89:
    html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N)     pretty_print(html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N))
Line 115: Line 115:
        else: print 'NaN'         else: print('NaN')
Line 140: Line 140:
    if start < 1 or end <=start: print "invalid start or end value"
    if n > end: print "WARNING: n is larger than the end value"		     if start < 1 or end <=start: print("invalid start or end value")
    if n > end: print("WARNING: n is larger than the end value")
Line 172: Line 172:
    #print x_cord, y_cord
Line 183: Line 182:
        #print x, "=x y=", y, " num =", num
Line 209: Line 207:
        print '(to go from x,y coords to an n, reset by setting n=0)'         print('(to go from x,y coords to an n, reset by setting n=0)')
Line 211: Line 209:
        #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
    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)
Line 233: Line 230:

Needs fix for show_factors
Line 239: Line 238:
    if start < 1 or end <=start: print "invalid start or end value"
    if n > end: print "WARNING: n is greater than end value"		     if start < 1 or end <=start: print("invalid start or end value")
    if n > end: print("WARNING: n is greater than end value")
Line 262: Line 261:
        print 'n =', factor(n)         print('n = {}'.format(factor(n)))
Line 284: Line 283:
            print 'Pink Curve: n^2 +', c
            print 'Green Curve: n^2 + n +', c2		             print('Pink Curve: n^2 +', c)
            print('Green Curve: n^2 + n +', c2)
Line 312: Line 311:
    print M; print '\n'*3
    print "Computing basis...\n\n"		     print(M)
    print('\n' * 3)
    print("Computing basis...\n\n")
Line 315: Line 315:
         print "Space has dimension 0"          print("Space has dimension 0")
Line 317: Line 317:
        prec = max(prec, M.dimension()+1)         prec = max(prec, M.dimension() + 1)
Line 320: Line 320:
    print "\n\n\nDone computing basis."     print("\n\n\nDone computing basis.")
Line 328: Line 328:
{{{#!sagecell
html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>')
ncols not working
{{{#!sagecell
pretty_print(html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>'))
Line 333: Line 335:
    print A.cuspidal_subgroup()     print(A.cuspidal_subgroup())
Line 475: Line 477:
                MP += text('$\omega^2$',(i+.5,r-j-.5),rgbcolor='black')                 MP += text(r'$\omega^2$',(i+.5,r-j-.5),rgbcolor='black')
Line 477: Line 479:
                MP += text('$\omega $',(i+.5,r-j-.5),rgbcolor='black')                 MP += text(r'$\omega $',(i+.5,r-j-.5),rgbcolor='black')
Line 486: Line 488:
    MP += text('$ \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]$.'%r)		     MP += text(r'$ \pi_1$',(r/2,r+2), rgbcolor='black', fontsize=25)
    MP += text(r'$ \pi_2$',(-2.5,r/2), rgbcolor='black', fontsize=25)

    pretty_print(html('Symmetry of Primary Cubic Residues mod ' \
          + r'%d primary primes in $ \mathbf Z[\omega]$.'%r))
Line 644: Line 646:
        html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js)))         pretty_print(html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js))))
Line 664: Line 666:
    html(s)}}}     pretty_print(html(s))
}}}
Line 754: Line 757:
    print "p = %s"%p
    show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0)		     print("p = %s" % p)
    show(E.change_ring(GF(p)).plot(), xmin=0, ymin=0)
Line 817: Line 820:
    c = list(continued_fraction(RealField(prec)(number))); print c     c = list(continued_fraction(RealField(prec)(number))); print(c)
Line 846: Line 849:
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)]		 L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in range(1000, -1, -1)]
R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in range(1000)]
xes = [x/1000.0 for x in range(-500,501,1)]
Line 872: Line 875:
= Multiple Zeta Values = = Multiple Zeta Values  =
Line 874: Line 877:
== Computing Multiple Zeta values == == Computing Multiple Zeta values  ==
Line 879: Line 882:
def _( weight=(5,(2..20))): def _( weight=(5,(2..100))):
Line 932: Line 935:
  print zeta(a)   u=zeta(a)
  RIF=RealIntervalField(int(3.321928*D))
  u=u/1
  print(u)
Line 939: Line 945:
def _( Depth=(5,(2..20))): def _( Depth=(5,(2..100))):
Line 998: Line 1004:

  print zeta(a)		   u=zeta(a)
  RIF=RealIntervalField(int(3.321928*D))
  u=u/1
  print(u)
Line 1002: Line 1010:
== Program to Compute Integer Relation between Multiple Zeta Values ==
{{{#!sagecell
from mpmath import *
print("Enter the number of composition")
@interact
def _( n=(5,(2..100))):
 a=[]
 for i in range(n):
        a.append([i+2,1])
 print("In each box Enter composition as an array")
 @interact
 def _(v=('Compositions', input_box( default=a, to_value=lambda x: vector(flatten(x)))), accuracy=(100..100000)):
  D=accuracy
  R=RealField(10)
  a=v
  def comptobin(a):
        word=[]
        for i in range(len(a)):
                word=word+[0]*(a[i]-1)+[1]
        return(word)
  DD=int(D)+int(R(log(3.321928*D))/R(log(10)))+4
  RIF=RealIntervalField(DD)
  mp.dps=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(mpf('0'))
                L.append(mpf('0'))
                if(word[i]==1 and i<k-1):
                        S.append(mpf('0'))
                        count=count+1
        T=mpf('1')
        for m in range(n):
                T=T/2
                B[k-1]=mpf('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=mpf('0')
        for i in range(len(l1)):
                Z=Z+l1[i]*l2[len(a)-i]
        return(Z)
  zet=[]
  for i in range(n):
        zet.append((zeta(comptobin(a[i]))))
  mp.dps=D
  for i in range(n):
        zet[i]=zet[i]/1
        print("zeta(", a[i], ")=", zet[i])
  u=pslq(zet,tol=10**-D,maxcoeff=100,maxsteps=10000)
  print("the Intger Relation between the above zeta values given by the vector")
  print(u)
}}}
{{attachment:akhi10.png}}
Line 1006: Line 1091:
def _( weight=(7,(2..30))): def _( weight=(7,(2..100))):
Line 1023: Line 1108:
  print "Composition is ",bintocomp(a)   print("Composition is {}".format(bintocomp(a)))
Line 1030: Line 1115:
def _( Depth=(7,(1..30))): def _( Depth=(7,(1..100))):
Line 1044: Line 1129:
  print "Word is  ",comptobin(a)   print("Word is {}".format(comptobin(a)))
Line 1051: Line 1136:
def _( weight=(7,(2..30))): def _( weight=(7,(2..100))):
Line 1066: Line 1151:
  print "Dual word is ",dual(a)   print("Dual word is {}"?format(dual(a)))
Line 1075: Line 1160:
def _( w1=(2,(2..20)), w2=(2,(2..20))): def _( w1=(2,(2..100)), w2=(2,(2..100))):
Line 1153: Line 1238:
    print c[1][i],"*",c[0][i] ,"+ ",
  print c[1][len(c[0])-1],"*",c[0][len(c[0])-1]		     print(c[1][i],"*",c[0][i] ,"+ ")
  print(c[1][len(c[0])-1],"*",c[0][len(c[0])-1])
Line 1162: Line 1247:
def _( w=(2,(2..20))): def _( w=(2,(2..100))):
Line 1302: Line 1387:
  c=Regshuf0(a)   c = Regshuf0(a)
Line 1305: Line 1390:
      print c[1][i],"*",c[0][i] ,"+ ",       print(c[1][i],"*",c[0][i] ,"+ ")
Line 1307: Line 1392:
    print c[1][len(c[0])-1],"*",c[0][len(c[0])-1]     print(c[1][len(c[0])-1],"*",c[0][len(c[0])-1])
Line 1455: Line 1540:
  c=Regshuf1(a)   c = Regshuf1(a)
Line 1458: Line 1543:
      print c[1][i],"*",c[0][i] ,"+ ",       print(c[1][i],"*",c[0][i] ,"+ ")
Line 1460: Line 1545:
    print c[1][len(c[0])-1],"*",c[0][len(c[0])-1]     print(c[1][len(c[0])-1],"*",c[0][len(c[0])-1])

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
    2. Computing Generalized Bernoulli Numbers
    3. Fundamental Domains of SL_2(ZZ)
  9. Multiple Zeta Values
    1. Computing Multiple Zeta values
      1. Word Input
      2. Composition Input
    2. Program to Compute Integer Relation between Multiple Zeta Values
    3. Word to composition
    4. Composition to Word
    5. Dual of a Word
    6. Shuffle product of two Words
    7. Shuffle Regularization at 0
    8. Shuffle Regularization at 1

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

Needs fix for show_factors

PolarSpiral.PNG

Modular Forms

Computing modular forms

by William Stein

modformbasis.png

Computing the cuspidal subgroup

by William Stein

ncols not working

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.

Computing Multiple Zeta values

Word Input

akhi1.png

Composition Input

akhi5.png

Program to Compute Integer Relation between Multiple Zeta Values

akhi10.png

Word to composition

akhi2.png

Composition to Word

akhi3.png

Dual of a Word

akhi4.png

Shuffle product of two Words

akhi6.png

Shuffle Regularization at 0

akhi7.png

Shuffle Regularization at 1

akhi8.png

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