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| = Sage Interactions - Number Theory = goto [:interact:interact main page] [[TableOfContents]] |
<<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}} |
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| {{{ | {{{#!sagecell |
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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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| 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') pretty_print(html(s)) }}} = Prime Numbers = == Illustrating the prime number theorem == |
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| {{{ @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 |
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| 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)) |
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| 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 == |
{{attachment:primes.png}} == Prime Spiral - Square FIXME == |
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| {{{ | {{{#!sagecell |
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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 else: print 'NaN' #Takes in an n and the start value of the spiral and gives its (x,y) coordinate |
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 |
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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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| 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") |
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| N = M.copy() | N = copy(M) |
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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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| #print x_cord, y_cord 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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| #print x, "=x y=", y, " num =", num | |
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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: print '(to go from x,y coords to an n, reset by setting n=0)' |
if n != 0: print('(to go from x,y coords to an n, reset by setting n=0)') |
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| #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) |
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| else: | else: |
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| attachment:SquareSpiral.PNG | {{attachment:SquareSpiral.PNG}} |
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| {{{ @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" |
Needs fix for show_factors {{{#!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") |
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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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| print 'n =', factor(n) |
print('n = {}'.format(factor(n))) |
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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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| print 'Pink Curve: n^2 +', c print 'Green Curve: n^2 + n +', c2 def g(m): return (a*m^2+b*m+c); |
print('Pink Curve: n^2 +', c) print('Green Curve: n^2 + n +', c2) def g(m): return (a*m^2+b*m+c); |
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| c= c2; | c= c2; |
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| attachment:PolarSpiral.PNG == Quadratic Residue Table == |
{{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 ncols not working {{{#!sagecell pretty_print(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 == |
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| {{{ | {{{#!sagecell |
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| attachment:quadres.png attachment:quadresbig.png == Cubic Residue Table == |
{{attachment:quadres.png}} {{attachment:quadresbig.png}} == 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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| MP += line([(i,0),(i,r)], rgbcolor='black') | MP += line([(i,0),(i,r)], rgbcolor='black') |
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| MP += text('$\omega^2$',(i+.5,r-j-.5),rgbcolor='black') | MP += text(r'$\omega^2$',(i+.5,r-j-.5),rgbcolor='black') |
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| MP += text('$\omega $',(i+.5,r-j-.5),rgbcolor='black') | MP += text(r'$\omega $',(i+.5,r-j-.5),rgbcolor='black') |
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| MP += text('$ \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]$.'%r)) |
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| attachment:cubres.png | {{attachment:cubres.png}} = Cyclotomic Fields = |
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| {{{ | {{{#!sagecell |
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| S = circle((0,0),1,rgbcolor='yellow') \ + line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) \ + line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) \ + line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) \ + point(e_pt,pointsize=50, rgbcolor='red') \ + point(f_pt,pointsize=50, rgbcolor='blue') \ + point(ef_pt,pointsize=50,rgbcolor='purple') \ + point(f_gs_pt,pointsize=75, rgbcolor='black') \ + point(e_gs_pt,pointsize=75, rgbcolor='black') \ + point(ef_gs_pt,pointsize=75, rgbcolor='black') \ + point(js_pt,pointsize=100,rgbcolor='green') |
S = circle((0,0),1,rgbcolor='yellow') S += line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) S += line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) S += line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) S += point(e_pt,pointsize=50, rgbcolor='red') S += point(f_pt,pointsize=50, rgbcolor='blue') S += point(ef_pt,pointsize=50,rgbcolor='purple') S += point(f_gs_pt,pointsize=75, rgbcolor='black') S += point(e_gs_pt,pointsize=75, rgbcolor='black') S += point(ef_gs_pt,pointsize=75, rgbcolor='black') S += point(js_pt,pointsize=100,rgbcolor='green') |
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| S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), \ | S += text('$J(%s,%s) = %s$'%(latex2(e),latex2(f),latex(js)), |
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| attachment:jacobising.png | {{attachment:jacobising.png}} |
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| {{{ | {{{#!sagecell |
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| S = circle((0,0),1,rgbcolor='yellow') \ + line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) \ + line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) \ + line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) \ + point(e_pt,pointsize=50, rgbcolor='red') \ + point(f_pt,pointsize=50, rgbcolor='blue') \ + point(ef_pt,pointsize=50,rgbcolor='purple') \ + point(f_gs_pt,pointsize=75, rgbcolor='black') \ + point(e_gs_pt,pointsize=75, rgbcolor='black') \ + point(ef_gs_pt,pointsize=75, rgbcolor='black') \ + point(js_pt,pointsize=100,rgbcolor='green') |
S = circle((0,0),1,rgbcolor='yellow') S += line([e_pt,e_gs_pt], rgbcolor='red', thickness=4) S += line([f_pt,f_gs_pt], rgbcolor='blue', thickness=3) S += line([ef_pt,ef_gs_pt], rgbcolor='purple',thickness=2) S += point(e_pt,pointsize=50, rgbcolor='red') S += point(f_pt,pointsize=50, rgbcolor='blue') S += point(ef_pt,pointsize=50,rgbcolor='purple') S += point(f_gs_pt,pointsize=75, rgbcolor='black') S += point(e_gs_pt,pointsize=75, rgbcolor='black') S += point(ef_gs_pt,pointsize=75, rgbcolor='black') S += point(js_pt,pointsize=100,rgbcolor='green') |
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| S += text('$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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| html('$$J(%s,%s) = %s$$'%(latex2(e),latex2(f),latex(js))) | pretty_print(html('$$J(%s,%s) = %s$$'%(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>') }}} attachment:jacobiexh.png |
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>' pretty_print(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 == by William Stein (Sage-2.10.3) {{{#!sagecell @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 {{{#!sagecell 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)] 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'], buttons=True,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}} = Multiple Zeta Values = by Akhilesh P. == Computing Multiple Zeta values == === Word Input === {{{#!sagecell R=RealField(10) @interact def _( weight=(5,(2..100))): 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) u=zeta(a) RIF=RealIntervalField(int(3.321928*D)) u=u/1 print(u) }}} {{attachment:akhi1.png}} === Composition Input === {{{#!sagecell R=RealField(10) @interact def _( Depth=(5,(2..100))): 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) u=zeta(a) RIF=RealIntervalField(int(3.321928*D)) u=u/1 print(u) }}} {{attachment:akhi5.png}} == 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}} == Word to composition == {{{#!sagecell @interact def _( weight=(7,(2..100))): 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 {}".format(bintocomp(a))) }}} {{attachment:akhi2.png}} == Composition to Word == {{{#!sagecell @interact def _( Depth=(7,(1..100))): 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 {}".format(comptobin(a))) }}} {{attachment:akhi3.png}} == Dual of a Word == {{{#!sagecell @interact def _( weight=(7,(2..100))): 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 dual(a): b=list() b=a b=b[::-1] for i in range(len(b)): b[i]=1-b[i] return(b) print("Dual word is {}"?format(dual(a))) }}} {{attachment:akhi4.png}} == Shuffle product of two Words == {{{#!sagecell @interact def _( w1=(2,(2..100)), w2=(2,(2..100))): a=[0] b=[0 for i in range(w2-1)] a=a+[1 for i in range(1,w1)] b=b+[1] import itertools #this program gives the list of all binary words of weight n and depth k @interact def _(v1=('word1', input_grid(1, w1, default=[a], to_value=lambda x: vector(flatten(x)))), v2=('word2', input_grid(1, w2, default=[b], to_value=lambda x: vector(flatten(x))))): a=[v1[i] for i in range(len(v1))] b=[v2[i] for i in range(len(v2))] def kbits(n, k): result = [] for bits in itertools.combinations(range(n), k): s = ['0'] * n for bit in bits: s[bit] = '1' result.append(''.join(s)) return result def sort(a,l,m): b=[] n=len(a) for i in range(n): b.append(a[i]) for j in range(l-1,-1,-1): k=0 for t in range(m+1): for i in range(n): if(a[i][j]== t): b[k]=a[i] k=k+1 for i in range(n): a[i]=b[i] return(a) def count(a): n=len(a) b=[] b.append(a[0]) m=[] m.append(1) c=0 for i in range(1,n): if(a[i]==a[i-1]): m[c]=m[c]+1 else: b.append(a[i]) m.append(1) c=c+1 return(b,m) def shuffle(a,b): r=len(a) s=len(b) # Generating an array of strings containing all combinations of weight r+s and depth s M=kbits(r+s,s) n=len(M) a1= [] for i in range(n): a1.append(list(M[i])) # The zeroes are replaced by the entries of a and the ones by the entries of b a2= [] for i in range(n): a2.append([]) count0=0 count1=0 for j in range(s+r): if(a1[i][j]=='0'): a2[i].append(a[count0]) count0=count0+1 if(a1[i][j]=='1'): a2[i].append(b[count1]) count1=count1+1 # Reordering in lexicographic order the entries of a2: this is done by first reordering them according to the last digit, then the next to last digit, etc a3=sort(a2,r+s,max(a+b+[0])) # Getting the same list without repetitions and with multiplicities a4=count(a3) return(a4) c=shuffle(a,b) for i in range(len(c[0])-1): print(c[1][i],"*",c[0][i] ,"+ ") print(c[1][len(c[0])-1],"*",c[0][len(c[0])-1]) }}} {{attachment:akhi6.png}} == Shuffle Regularization at 0 == {{{#!sagecell @interact def _( w=(2,(2..100))): a=[0] a=a+[1 for i in range(1,w)] import itertools #this program gives the list of all binary words of weight n and depth k @interact def _(v=('word', input_grid(1, w, default=[a], to_value=lambda x: vector(flatten(x))))): a=[v[i] for i in range(len(v))] def kbits(n, k): result = [] for bits in itertools.combinations(range(n), k): s = ['0'] * n for bit in bits: s[bit] = '1' result.append(''.join(s)) return result def sort(a,l,m): b=[] n=len(a) for i in range(n): b.append(a[i]) for j in range(l-1,-1,-1): k=0 for t in range(m+1): for i in range(n): if(a[i][j]== t): b[k]=a[i] k=k+1 for i in range(n): a[i]=b[i] return(a) def sort1(a,l,m): b=[] b.append([]) b.append([]) n=len(a[0]) for i in range(n): b[0].append(a[0][i]) b[1].append(a[1][i]) for j in range(l-1,-1,-1): k=0 for t in range(m+1): for i in range(n): if(a[0][i][j]== t): b[0][k]=a[0][i] b[1][k]=a[1][i] k=k+1 for i in range(n): a[0][i]=b[0][i] a[1][i]=b[1][i] return(a) def count(a): n=len(a) b=[] b.append(a[0]) m=[] m.append(1) c=0 for i in range(1,n): if(a[i]==a[i-1]): m[c]=m[c]+1 else: b.append(a[i]) m.append(1) c=c+1 return(b,m) def count1(a): n=len(a[0]) b=[] b.append([]) b.append([]) b[0].append(a[0][0]) b[1].append(a[1][0]) c=0 for i in range(1,n): if(a[0][i]==a[0][i-1]): b[1][c]=b[1][c]+a[1][i] else: b[0].append(a[0][i]) b[1].append(a[1][i]) c=c+1 return(b) def shuffle(a,b): r=len(a) s=len(b) # Generating an array of strings containing all combinations of weight r+s and depth s M=kbits(r+s,s) n=len(M) a1= [] for i in range(n): a1.append(list(M[i])) # The zeroes are replaced by the entries of a and the ones by the entries of b a2= [] for i in range(n): a2.append([]) count0=0 count1=0 for j in range(s+r): if(a1[i][j]=='0'): a2[i].append(a[count0]) count0=count0+1 if(a1[i][j]=='1'): a2[i].append(b[count1]) count1=count1+1 # Reordering in lexicographic order the entries of a2: this is done by first reordering them according to the last digit, then the next to last digit, etc a3=sort(a2,r+s,max(a+b+[0])) # Getting the same list without repetitions and with multiplicities a4=count(a3) return(a4) def Regshuf0(a): r=[] r.append([]) r.append([]) t=0 c=1 for i in range(len(a)+1): if(t==0): b=shuffle(a[:len(a)-i],a[len(a)-i:]) for j in range(len(b[0])): r[0].append(b[0][j]) r[1].append(b[1][j]*c) c=-c if(i<len(a)): if(a[len(a)-1-i]==1): t=1 r=sort1(r,len(a),max(a+[0])) r=count1(r) rg=[] rg.append([]) rg.append([]) for i in range(len(r[0])): if(r[1][i] is not 0): rg[0].append(r[0][i]) rg[1].append(r[1][i]) return(rg) c = Regshuf0(a) for i in range(len(c[0])-1): if(c[1][i] != 0): print(c[1][i],"*",c[0][i] ,"+ ") if(c[1][len(c[0])-1] != 0): print(c[1][len(c[0])-1],"*",c[0][len(c[0])-1]) }}} {{attachment:akhi7.png}} == Shuffle Regularization at 1 == {{{#!sagecell @interact def _( w=(2,(2..20))): a=[0] a=a+[1 for i in range(1,w)] import itertools #this program gives the list of all binary words of weight n and depth k @interact def _(v=('word', input_grid(1, w, default=[a], to_value=lambda x: vector(flatten(x))))): a=[v[i] for i in range(len(v))] def kbits(n, k): result = [] for bits in itertools.combinations(range(n), k): s = ['0'] * n for bit in bits: s[bit] = '1' result.append(''.join(s)) return result def sort(a,l,m): b=[] n=len(a) for i in range(n): b.append(a[i]) for j in range(l-1,-1,-1): k=0 for t in range(m+1): for i in range(n): if(a[i][j]== t): b[k]=a[i] k=k+1 for i in range(n): a[i]=b[i] return(a) def sort1(a,l,m): b=[] b.append([]) b.append([]) n=len(a[0]) for i in range(n): b[0].append(a[0][i]) b[1].append(a[1][i]) for j in range(l-1,-1,-1): k=0 for t in range(m+1): for i in range(n): if(a[0][i][j]== t): b[0][k]=a[0][i] b[1][k]=a[1][i] k=k+1 for i in range(n): a[0][i]=b[0][i] a[1][i]=b[1][i] return(a) def count(a): n=len(a) b=[] b.append(a[0]) m=[] m.append(1) c=0 for i in range(1,n): if(a[i]==a[i-1]): m[c]=m[c]+1 else: b.append(a[i]) m.append(1) c=c+1 return(b,m) def count1(a): n=len(a[0]) b=[] b.append([]) b.append([]) b[0].append(a[0][0]) b[1].append(a[1][0]) c=0 for i in range(1,n): if(a[0][i]==a[0][i-1]): b[1][c]=b[1][c]+a[1][i] else: b[0].append(a[0][i]) b[1].append(a[1][i]) c=c+1 return(b) def shuffle(a,b): r=len(a) s=len(b) # Generating an array of strings containing all combinations of weight r+s and depth s M=kbits(r+s,s) n=len(M) a1= [] for i in range(n): a1.append(list(M[i])) # The zeroes are replaced by the entries of a and the ones by the entries of b a2= [] for i in range(n): a2.append([]) count0=0 count1=0 for j in range(s+r): if(a1[i][j]=='0'): a2[i].append(a[count0]) count0=count0+1 if(a1[i][j]=='1'): a2[i].append(b[count1]) count1=count1+1 # Reordering in lexicographic order the entries of a2: this is done by first reordering them according to the last digit, then the next to last digit, etc a3=sort(a2,r+s,max(a+b+[0])) # Getting the same list without repetitions and with multiplicities a4=count(a3) return(a4) def Regshuf1(a): r=[] r.append([]) r.append([]) t=0 c=1 for i in range(len(a)+1): if(t==0): b=shuffle(a[:i],a[i:]) for j in range(len(b[0])): r[0].append(b[0][j]) r[1].append(b[1][j]*c) c=-c if(i<len(a)): if(a[i]==0): t=1 r=sort1(r,len(a),max(a+[0])) r=count1(r) rg=[] rg.append([]) rg.append([]) for i in range(len(r[0])): if(r[1][i] is not 0): rg[0].append(r[0][i]) rg[1].append(r[1][i]) return(rg) c = Regshuf1(a) for i in range(len(c[0])-1): if(c[1][i] != 0): print(c[1][i],"*",c[0][i] ,"+ ") if(c[1][len(c[0])-1] != 0): print(c[1][len(c[0])-1],"*",c[0][len(c[0])-1]) }}} {{attachment:akhi8.png}} |
Contents
Integer Factorization
Divisibility Poset
by William Stein
Factor Trees
by William Stein
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
Prime Spiral - Square FIXME
by David Runde
Prime Spiral - Polar
by David Runde
Needs fix for show_factors
Modular Forms
Computing modular forms
by William Stein
Computing the cuspidal subgroup
by William Stein
ncols not working
A Charpoly and Hecke Operator Graph
by William Stein
Modular Arithmetic
Quadratic Residue Table FIXME
by Emily Kirkman
Cubic Residue Table FIXME
by Emily Kirkman
Cyclotomic Fields
Gauss and Jacobi Sums in Complex Plane
by Emily Kirkman
Exhaustive Jacobi Plotter
by Emily Kirkman
Elliptic Curves
Adding points on an elliptic curve
by David Møller Hansen
Plotting an elliptic curve over a finite field
Cryptography
The Diffie-Hellman Key Exchange Protocol
by Timothy Clemans and William Stein
Other
Continued Fraction Plotter
by William Stein
Computing Generalized Bernoulli Numbers
by William Stein (Sage-2.10.3)
Fundamental Domains of SL_2(ZZ)
by Robert Miller
Multiple Zeta Values
by Akhilesh P.
Computing Multiple Zeta values
Word Input
Composition Input
Program to Compute Integer Relation between Multiple Zeta Values
Word to composition
Composition to Word
Dual of a Word
Shuffle product of two Words
Shuffle Regularization at 0
Shuffle Regularization at 1
