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2*10
20
parametric_plot((sin(3*x),cos(5*x)),(x,0,2*pi))
def pair(x): if x%2==0: return True return False def syracuse(x): if pair(x): return x//2 else: return 3*x+1 def suite_de_syracuse(u,n): for i in range(1,n): u=syracuse(u) print(u) return u def suite_de_syracuse_1(u): n=0 while u!=1: n+=1 u=syracuse(u) return n suite_de_syracuse_1(1000)
111
var('x,y,z') f=(x+y+z)^4 fg=f.expand() print(fg)
(x, y, z) x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + y^4 + 4*x^3*z + 12*x^2*y*z + 12*x*y^2*z + 4*y^3*z + 6*x^2*z^2 + 12*x*y*z^2 + 6*y^2*z^2 + 4*x*z^3 + 4*y*z^3 + z^4
f.coefficient(x^2*y^2)
0
p=3*x^2*y+7*x*y^2+y^3 p.coefficient(x^2*y)
3
fg1=str(fg).split(' + ') for d in fg1: r=1 try: int(d[0]) r=int(d.split('*')[0]) except: pass print(r)
1 4 4 6 6 4 4 1 4 4 1 12 1 12 4 4 6 6 1 12 6 6 4 4 4 4 1
def nb_chif_coef_dev(u): var('x,y,z') n=0 while True: n+=1 f=(x+y+z)^n fd=str(f.expand()).split(' + ') for d in fd: try: int(d[0]) if len(d.split('*')[0])>u-1: return n except: pass nb_chif_coef_dev(30)
65
def champernowne(): c='0.' n=0 while True: n+=1 c+=str(n) if '888' in c: return c.find('888')-1 champernowne()
166
g='1235689478' g[-3:]
'478'
def test_euler(n,p): if ((n^((p-1)/2)-1)/p).is_integral() or ((n^((p-1)/2)+1)/p).is_integral(): return True return False def enigme_euler(p): te = False for n in (2,3,5): if test_euler(n,p): te=True else: te=False break if te and not is_prime(p): return True return False p=3 #enigme_euler(1729) while not enigme_euler(p): p+=2 print(p)
1729
-1%217 ((2^((217-1)/2)-1)/217).is_integral()
216 True
var('x,a') f=x^2-x-1 r=solve(f,x) print(str(r[1]).split('==')[1]) t=((1+sqrt(5))/2)^(2*a/pi) #t2=((1+sqrt(5))/2)^(2*a/pi) print(t) polar_plot(t,(a,0,100*pi))
(x, a) 1/2*sqrt(5) + 1/2 (1/2*sqrt(5) + 1/2)^(2*a/pi)
help(polar_plot)
Help on function polar_plot in module sage.plot.plot: polar_plot(*args, **kwds) ``polar_plot`` takes a single function or a list or tuple of functions and plots them with polar coordinates in the given domain. This function is equivalent to the :func:`plot` command with the options ``polar=True`` and ``aspect_ratio=1``. For more help on options, see the documentation for :func:`plot`. INPUT: - ``funcs`` - a function - other options are passed to plot EXAMPLES: Here is a blue 8-leaved petal:: sage: polar_plot(sin(5*x)^2, (x, 0, 2*pi), color='blue') Graphics object consisting of 1 graphics primitive .. PLOT:: g = polar_plot(sin(5*x)**2, (x, 0, 2*pi), color='blue') sphinx_plot(g) A red figure-8:: sage: polar_plot(abs(sqrt(1 - sin(x)^2)), (x, 0, 2*pi), color='red') Graphics object consisting of 1 graphics primitive .. PLOT:: g = polar_plot(abs(sqrt(1 - sin(x)**2)), (x, 0, 2*pi), color='red') sphinx_plot(g) A green limacon of Pascal:: sage: polar_plot(2 + 2*cos(x), (x, 0, 2*pi), color=hue(0.3)) Graphics object consisting of 1 graphics primitive .. PLOT:: g = polar_plot(2 + 2*cos(x), (x, 0, 2*pi), color=hue(0.3)) sphinx_plot(g) Several polar plots:: sage: polar_plot([2*sin(x), 2*cos(x)], (x, 0, 2*pi)) Graphics object consisting of 2 graphics primitives .. PLOT:: g = polar_plot([2*sin(x), 2*cos(x)], (x, 0, 2*pi)) sphinx_plot(g) A filled spiral:: sage: polar_plot(sqrt, 0, 2 * pi, fill=True) Graphics object consisting of 2 graphics primitives .. PLOT:: g = polar_plot(sqrt, 0, 2 * pi, fill=True) sphinx_plot(g) Fill the area between two functions:: sage: polar_plot(cos(4*x) + 1.5, 0, 2*pi, fill=0.5 * cos(4*x) + 2.5, fillcolor='orange') Graphics object consisting of 2 graphics primitives .. PLOT:: g = polar_plot(cos(4*x) + 1.5, 0, 2*pi, fill=0.5 * cos(4*x) + 2.5, fillcolor='orange') sphinx_plot(g) Fill the area between several spirals:: sage: polar_plot([(1.2+k*0.2)*log(x) for k in range(6)], 1, 3 * pi, fill={0: [1], 2: [3], 4: [5]}) Graphics object consisting of 9 graphics primitives .. PLOT:: g = polar_plot([(1.2+k*0.2)*log(x) for k in range(6)], 1, 3 * pi, fill={0: [1], 2: [3], 4: [5]}) sphinx_plot(g) Exclude points at discontinuities:: sage: polar_plot(log(floor(x)), (x, 1, 4*pi), exclude=[1..12]) Graphics object consisting of 12 graphics primitives .. PLOT:: g = polar_plot(log(floor(x)), (x, 1, 4*pi), exclude=list(range(1,13))) sphinx_plot(g)