scratch paper.sagews

mod(31^(-1),192)

31

legendre_symbol(118,131)

-1

[r]+[mod(r^c,m) for c in [1..m]]
mod(31^(-1),192)

Error in lines 1-1 Traceback (most recent call last): File "/projects/cec84faa-0c9f-4d53-a7fe-4962a22dc313/.sagemathcloud/sage_server.py", line 879, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'm' is not defined

m=7
headerRow=['x']+range(m)
data=[headerRow]
for r in range(m):
    data.append([r]+[mod(r*c,m) for c in range(m)])
table(data,header_row=True, header_column=True)

x | 0 1 2 3 4 5 6 +---+---+---+---+---+---+---+---+ 0 | 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 2 | 0 2 4 6 1 3 5 3 | 0 3 6 2 5 1 4 4 | 0 4 1 5 2 6 3 5 | 0 5 3 1 6 4 2 6 | 0 6 5 4 3 2 1

m=7
headerRow=['+']+range(m)
data=[headerRow]
for r in range(m):
    data.append([r]+[mod(r+c,m) for c in range(m)])
table(data,header_row=True, header_column=True)

+ | 0 1 2 3 4 5 6 +---+---+---+---+---+---+---+---+ 0 | 0 1 2 3 4 5 6 1 | 1 2 3 4 5 6 0 2 | 2 3 4 5 6 0 1 3 | 3 4 5 6 0 1 2 4 | 4 5 6 0 1 2 3 5 | 5 6 0 1 2 3 4 6 | 6 0 1 2 3 4 5

m=7
headerRow=['^']+[1..m-1]
data=[headerRow]
for r in [1..m-1]:
    data.append([r]+[mod(r^c,m) for c in [1..m-1]])
table(data,header_row=True, header_column=True)

^ | 1 2 3 4 5 6 +---+---+---+---+---+---+---+ 1 | 1 1 1 1 1 1 2 | 2 4 1 2 4 1 3 | 3 2 6 4 5 1 4 | 4 2 1 4 2 1 5 | 5 4 6 2 3 1 6 | 6 1 6 1 6 1

m=17
headerRow=['^']+[1..m-1]
data=[headerRow]
for r in [1..m-1]:
    data.append([r]+[mod(r^c,m) for c in [1..m-1]])
table(data,header_row=True, header_column=True)

^ | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 +----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+ 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 | 2 4 8 16 15 13 9 1 2 4 8 16 15 13 9 1 3 | 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6 1 4 | 4 16 13 1 4 16 13 1 4 16 13 1 4 16 13 1 5 | 5 8 6 13 14 2 10 16 12 9 11 4 3 15 7 1 6 | 6 2 12 4 7 8 14 16 11 15 5 13 10 9 3 1 7 | 7 15 3 4 11 9 12 16 10 2 14 13 6 8 5 1 8 | 8 13 2 16 9 4 15 1 8 13 2 16 9 4 15 1 9 | 9 13 15 16 8 4 2 1 9 13 15 16 8 4 2 1 10 | 10 15 14 4 6 9 5 16 7 2 3 13 11 8 12 1 11 | 11 2 5 4 10 8 3 16 6 15 12 13 7 9 14 1 12 | 12 8 11 13 3 2 7 16 5 9 6 4 14 15 10 1 13 | 13 16 4 1 13 16 4 1 13 16 4 1 13 16 4 1 14 | 14 9 7 13 12 15 6 16 3 8 10 4 5 2 11 1 15 | 15 4 9 16 2 13 8 1 15 4 9 16 2 13 8 1 16 | 16 1 16 1 16 1 16 1 16 1 16 1 16 1 16 1

m=11
headerRow=['^']+[1..m-1]
data=[headerRow]
for r in [1..m-1]:
    data.append([r]+[mod(r^c,m) for c in [1..m-1]])
table(data,header_row=True, header_column=True)

^ | 1 2 3 4 5 6 7 8 9 10 +----+----+---+----+---+----+---+----+---+----+----+ 1 | 1 1 1 1 1 1 1 1 1 1 2 | 2 4 8 5 10 9 7 3 6 1 3 | 3 9 5 4 1 3 9 5 4 1 4 | 4 5 9 3 1 4 5 9 3 1 5 | 5 3 4 9 1 5 3 4 9 1 6 | 6 3 7 9 10 5 8 4 2 1 7 | 7 5 2 3 10 4 6 9 8 1 8 | 8 9 6 4 10 3 2 5 7 1 9 | 9 4 3 5 1 9 4 3 5 1 10 | 10 1 10 1 10 1 10 1 10 1

len(prime_range(100))

prime_pi(10^2)

Mod(5, 12)

25 25 5

sum([1,2])

3

sum(prime_range(100))

1060

12.divides(4)

False

[1, 3, 5, 7, 9]

[n for n in [1..10] if n.divides(100)]

[1, 2, 4, 5, 10]

(100.divisors())^2
[n^2 for n in 100.divisors()]

[1, 4, 16, 25, 100, 400, 625, 2500, 10000]

P=Primes()algebr

P=Primes()

P

Set of all prime numbers: 2, 3, 5, 7, ...

P.next(9)

11

P.unrank(1)

3

P.unrank?

File: /usr/local/sage/sage-6.4/local/lib/python2.7/site-packages/sage/sets/primes.py
Signature : P.unrank()
Docstring :
Returns the n-th prime number.

EXAMPLES::
   sage: P = Primes() sage: P.unrank(0) 2 sage: P.unrank(5) 13
   sage: P.unrank(42) 191

n=100000
i=99900
while i<n:
    p=P.unrank(i)
    q= p+2
    if q.is_prime():
        print p,q
        i = i+1
    i =i+1

1298489 1298491 1298651 1298653 1298909 1298911 1299059 1299061 1299209 1299211 1299341 1299343 1299377 1299379 1299437 1299439 1299449 1299451

n.is_square()

False

n

100000

9.is_square()

True

def T(n):
    return sum([1..n])

T(4)

10

n=10000
for k in [1..n]:
    if T(k).is_square():
        print k, T(k), sqrt(T(k))

1 1 1 8 36 6 49 1225 35 288 41616 204 1681 1413721 1189 9800 48024900 6930

2013.factor()

3 * 11 * 61

gcd(11,144)

1

xgcd(11,144)

(1, -13, 1)

g,x,y = xgcd(11,144)

g

1

x

-13

# Press the TAB key after 'graphs.' to see a list of predefined graphs.
graphs.CompleteGraph(5).show(vertex_labels=False, vertex_size=0,edge_labels=True)

show(graphs.PetersenGraph())

d3-based renderer not yet implemented

g=graphs.CompleteGraph(7)
label=1
for u,v,l in g.edges():
    g.set_edge_label(u,v,str(label))
    label +=1
show(g,vertex_labels=false,edge_labels=True)

d3-based renderer not yet implemented

a

from sage.plot.colors import rainbow
C = graphs.CubeGraph(5)
R = rainbow(5)
edge_colors = {}
for i in [0..4]:
    edge_colors[R[i]] = []
for u,v,l in C.edges():
    for i in range(5):
        if u[i] != v[i]:
            edge_colors[R[i]].append((u,v,l))
C.graphplot(vertex_labels=False, vertex_size=0, edge_colors=edge_colors).show()

G = graphs.HeawoodGraph().copy(sparse=True)
for u,v,l in G.edges():
    G.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')')
G.graphplot(edge_labels=True).show()

x = 500000

prime_pi(x)

41538

def pnt(x):
    return float (x/log(x))

pnt(500000)

38102.89241501728

numerical_integral(1/log(x), 2,10)  # [1] gives error bound

(0.6096462786402765, 0.0)

def li(x):
    var('t')
    return numerical_integral(1/log(t), 2,x) 

li(500000)    

(41605.243622658, 0.0009425010476448864)

prime_pi(500000)

41538

pnt(1000)

144.76482730108395

prime_pi(1000)

168

li(1000)

(176.5644942100379, 1.6446995671159058e-06)

7*log(7.)

13.6213710433872

[1; 1, 1, 3, 31, 1, 145, 1, 4, 2, 8, 1, 6, 1, 2, 3, 1, 4, 1, 5, ...]

cp.convergents()[4]

103993/33102

latex(7+1/9)

\frac{64}{9}

$\pi = 3 +\frac{1}{7+\frac{1}{15}}$

$\pi=3+\cfrac{1}{7+ \cfrac{1}{15+ \cfrac{1}{1+\cfrac{1}{292\dotsb }}}}$

continued_fraction(e)

[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, ...]

float(pi/2)

1.5707963267948966

float(1./sqrt(2)+1./sqrt(3))

1.2844570503761732

float(sqrt(2)+sqrt(3))

3.1462643699419726

factor(2015)

5 * 13 * 31

prime_pi(10^6)

78498

for t in [1..7]:
    a=prime_pi(10^t)
    b=prime_pi(10^(t-1))
    print t, a,a-b, float(a/10^t)

1 4 4 0.4 2 25 21 0.25 3 168 143 0.168 4 1229 1061 0.1229 5 9592 8363 0.09592 6 78498 68906 0.078498 7 664579 586081 0.0664579

list = [prime_pi(2^t) for t in [1..18]]

list

[1, 2, 4, 6, 11, 18, 31, 54, 97, 172, 309, 564, 1028, 1900, 3512, 6542, 12251, 23000]

ratio=[float((list[t]-list[t-1])/(list[t-1]-list[t-2])) for t in [2..17]]

ratio

[2.0, 1.0, 2.5, 1.4, 1.8571428571428572, 1.7692307692307692, 1.8695652173913044, 1.744186046511628, 1.8266666666666667, 1.8613138686131387, 1.8196078431372549, 1.8793103448275863, 1.848623853211009, 1.879652605459057, 1.884158415841584, 1.882816605359958]

plot(1/log(x),(x,2,100))

P=Primes()
def prod_primes(l):
    prod=1
    for k in range(l):
        prod *= P.unrank(k)
    return prod

prod_primes(3)

30

for k in [1..30]:
    a = prod_primes(k)
    k, a, sigma(a),float(sigma(a)/a)

(1, 2, 3, 1.5) (2, 6, 12, 2.0) (3, 30, 72, 2.4) (4, 210, 576, 2.742857142857143) (5, 2310, 6912, 2.9922077922077923) (6, 30030, 96768, 3.2223776223776226) (7, 510510, 1741824, 3.4119292472233647) (8, 9699690, 34836480, 3.5915044707614365) (9, 223092870, 836075520, 3.7476568390554124) (10, 6469693230, 25082265600, 3.876886385229737) (11, 200560490130, 802632499200, 4.00194723636618) (12, 7420738134810, 30500034969600, 4.110107972484185) (13, 304250263527210, 1281001468723200, 4.210354508398433) (14, 13082761331670030, 56364064623820800, 4.308269729523978) (15, 614889782588491410, 2705475101943398400, 4.399935042918106) (16, 32589158477190044730, 146095655504943513600, 4.482952685237315) (17, 1922760350154212639070, 8765739330296610816000, 4.558934934139642) (18, 117288381359406970983270, 543475838478389870592000, 4.633671572404227) (19, 7858321551080267055879090, 36956357016530511200256000, 4.70283084960429) (20, 557940830126698960967415390, 2660857705190196806418432000, 4.769067903824069) (21, 40729680599249024150621323470, 196903470184074563674963968000, 4.834397601136727) (22, 3217644767340672907899084554130, 15752277614725965093997117440000, 4.89559250748023) (23, 267064515689275851355624017992790, 1323191319636981067895757864960000, 4.954575549739028) (24, 23768741896345550770650537601358310, 119087218767328296110618207846400000, 5.010244937938342) (25, 2305567963945518424753102147331756070, 11670547439198173018840584368947200000, 5.06189694760781) (26, 232862364358497360900063316880507363070, 1190395838798213647921739605632614400000, 5.112014739168283) (27, 23984823528925228172706521638692258396210, 123801167235014219383860918985791897600000, 5.16164595022817) (28, 2566376117594999414479597815340071648394470, 13370526061381535693456979250465524940800000, 5.209885632006003) (29, 279734996817854936178276161872067809674997230, 1470757866751968926280267717551207743488000000, 5.2576827478959665) (30, 31610054640417607788145206291543662493274686990, 167666396809724457595950519800837682757632000000, 5.30421091380655)

def h(n):
    prod = 1.
    for k in range(n):
        p = P.unrank(k)
        prod *= float((p+1)/p)
    return prod

h(1)

1.50000000000000

h(10)

3.87688638522974

h(100)

6.85163642192891

h(1000)

9.73215913491596

vals=[h(n) for n in [1..1000]]

list_plot(vals)

float(log(1000))

6.907755278982137

h(1000)-h(100)

2.88052271298704

h(100)-h(10)

2.97475003669918

h(10000)-h(1000)

2.78720754884904

163.is_prime()

True

def primeSum(n):
    sum =0
    for p in prime_range(n):
        sum += 1./p
    return sum

def primeSumList(n):
    list=[]
    sum =0
    for p in prime_range(n):
        sum += 1./p
        list.append(sum)
    return list

primeSumList(100)

[0.500000000000000, 0.833333333333333, 1.03333333333333, 1.17619047619048, 1.26709956709957, 1.34402264402264, 1.40284617343441, 1.45547775238178, 1.49895601325134, 1.53343877187203, 1.56569683638816, 1.59272386341519, 1.61711410731763, 1.64036992127112, 1.66164651701580, 1.68051444154410, 1.69746359408647, 1.71385703670942, 1.72878240984375, 1.74286691688600, 1.75656554702299, 1.76922377487109, 1.78127196764218, 1.79250792269836, 1.80281720104887]

n=1000
psl=list_plot(primeSumList(n),plotjoined=True)
lg = plot(log(x),(x,2,prime_pi(n)),color='red')
lg2 = plot(log(log(x)),(x,2,prime_pi(n)),color='green')
show(lg+psl+lg2)

n=10000000
psl=list_plot(primeSumList(n),plotjoined=True)
lg = plot(log(x),(x,2,prime_pi(n)),color='red')
lg2 = plot(log(log(x)),(x,2,prime_pi(n)),color='green')
show(psl+lg2)

n=3
l=100
sum=0
for k in [2..l]:
    sum += 1./k^n
sum

0.202007400659678

numerical_integral(1/x^3,1,100)[0]  # [1] gives error bound

0.49995000000000006

n=100
prod=1
for k in [1..n]:
    prod *= 1.+1./2^k
prod    

2.38423102903137

binomial(9,3)

84

n=1000
p=17
k=110
bc = binomial(n,k)

bc

124418211871840133364593075107752735363117851087126726877571151388277607346819388158474302733831496092331114788214489789460593749920645314041529598400

factor(bc)

2^6 * 3^5 * 5^2 * 11 * 19 * 23 * 29 * 31^2 * 37 * 41 * 43 * 47 * 61 * 71 * 83 * 113 * 131 * 137 * 139 * 149 * 151 * 157 * 163 * 179 * 181 * 191 * 193 * 197 * 199 * 223 * 227 * 229 * 233 * 239 * 241 * 307 * 311 * 313 * 317 * 331 * 449 * 457 * 461 * 463 * 467 * 479 * 487 * 491 * 499 * 907 * 911 * 919 * 929 * 937 * 941 * 947 * 953 * 967 * 971 * 977 * 983 * 991 * 997

def sq(n):
    return n^2

5.sq()

Error in lines 1-1 Traceback (most recent call last): File "/projects/cec84faa-0c9f-4d53-a7fe-4962a22dc313/.sagemathcloud/sage_server.py", line 879, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/structure/element.pyx", line 438, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4406) return getattr_from_other_class(self, P._abstract_element_class, name) File "sage/structure/misc.pyx", line 257, in sage.structure.misc.getattr_from_other_class (build/cythonized/sage/structure/misc.c:1631) raise dummy_attribute_error AttributeError: 'sage.rings.integer.Integer' object has no attribute 'sq'

sq(5)

25

def star(f,g,n):
    sum=0
    for d in n.divisors():
        sum += f(d)*g(n/d)
    return sum

def one(n):
    return 1

moebius(9)

0

euler_phi(9)

6

star(moebius,moebius,12)

-2

for n in [1..29]:
    print n, star(moebius, euler_phi,n)

1 1 2 0 3 1 4 1 5 3 6 0 7 5 8 2 9 4 10 0 11 9 12 1 13 11 14 0 15 3 16 4 17 15 18 0 19 17 20 3 21 5 22 0 23 21 24 2 25 16 26 0 27 12 28 5 29 27

c=.7
c2=.5
for k in [1..10]:
    n = 10^k
    lg=float(log(n))
    v1=c*n/lg-1
    v2=c2*n/lg
    print n,v1, v2

10 2.04006137332276 2.17147240951626 100 14.2003068666138 10.8573620475813 1000 100.335379110759 72.3824136505420 10000 759.015343330691 542.868102379065 100000 6079.12274664553 4342.94481903252 1000000 50666.6895553794 36191.2068252710 10000000 434293.481903252 310210.344216608 100000000 3.80007571665345e6 2.71434051189532e6 1000000000 3.37784587035863e7 2.41274712168473e7 10000000000 3.04006136332276e8 2.17147240951626e8

float(log(10,2))

3.3219280948873626

n=1000
end= floor(log(n,2))
sum = 0
for k in [1..end]:
    term = float(log(2)*(n/2^k)/log((n/2^k)))
    sum += term
    print term
print sum

55.7675696989 31.3842292317 17.9448565864 10.4764096417 6.29306433961 3.93995206961 2.63421046514 1.9871203725 2.02232012989 132.449732535

prime_pi(100)

25

star(euler_phi,moebius,17)

15

list_plot([euler_phi(n) for n in [1..1000]],aspect_ratio=1)

P=Primes()

P

Set of all prime numbers: 2, 3, 5, 7, ...

n=100
primes=prime_range(3,n)
count1=0
count3=0
for p in primes:
    if mod(p,4)==1:
        count1 +=1
    else:
        count3 +=1
    print p,count1, count3
    

3 0 1 5 1 1 7 1 2 11 1 3 13 2 3 17 3 3 19 3 4 23 3 5 29 4 5 31 4 6 37 5 6 41 6 6 43 6 7 47 6 8 53 7 8 59 7 9 61 8 9 67 8 10 71 8 11 73 9 11 79 9 12 83 9 13 89 10 13 97 11 13

#density example
#A is a subset of the natural numbers
#we want to estimate d(A)

#example where A is the even integers
 
n = 1000 #this can be changed
count = 0
for k in [1..n]:
    if mod(k,2) ==0:
        count += 1
print float(count/n)

0.5

P = Primes()
len = 5
tests= 1000
for _ in range(tests):
    d = randint(3,100)
    start = P.unrank(randint(1,100))
    prod=1
    for k in range(len):
        prod *= (start + d*k).is_prime()
    if prod ==1: 
        print start, d
    

103 90 277 30

439.is_prime()

True

for m in [1..1000]:
    if mod(11^(m-1),m)==1:
        if m.is_prime()==False:
            print m, factor(m)

1 1 10 2 * 5 15 3 * 5 70 2 * 5 * 7 133 7 * 19 190 2 * 5 * 19 259 7 * 37 305 5 * 61 481 13 * 37 645 3 * 5 * 43 703 19 * 37 793 13 * 61

mod(2^340,341)

1

341.is_prime()

False

n=10000
for x in [1..n]:
    y=floor(1.4*x)
    while y^2-2*x^2<=1:
        if y^2-2*x^2==1:
            print x,y
        y += 1

2 3 12 17 70 99 408 577 2378 3363

a=[[1,1],[2,1]]

u=exp(Matrix(a))

latex(u)

\left(\begin{array}{rr} \frac{1}{2} \, {\left(e^{\left(2 \, \sqrt{2}\right)} + 1\right)} e^{\left(-\sqrt{2} + 1\right)} & \frac{1}{4} \, {\left(\sqrt{2} e^{\left(2 \, \sqrt{2}\right)} - \sqrt{2}\right)} e^{\left(-\sqrt{2} + 1\right)} \\ \frac{1}{2} \, {\left(\sqrt{2} e^{\left(2 \, \sqrt{2}\right)} - \sqrt{2}\right)} e^{\left(-\sqrt{2} + 1\right)} & \frac{1}{2} \, {\left(e^{\left(2 \, \sqrt{2}\right)} + 1\right)} e^{\left(-\sqrt{2} + 1\right)} \end{array}\right)

n=1000
sum =0
sumlist=[]
for p in prime_range(n):
    sum += float(1/p)
    sumlist.append(sum)

sumlist

[0.5, 0.8333333333333333, 1.0333333333333332, 1.176190476190476]

pp=list_plot(zip(prime_range(n),sumlist),plotjoined=true)

pl=plot(log(log(x)),(x,2,n),color='red')
pl1=plot(log(x),(x,2,n),color='green')

n=1000
sum =0
harmlist=[]
for k in [1..n]:
    sum +=float(1/k)
    harmlist.append(sum)

hp=list_plot(zip([1..n],harmlist),plotjoined=true,color='black')

show(pp+pl+pl1+hp)

n=1000
sum =0
wharmlist=[]
for k in [2..n]:
    sum +=float(1/(log(k)*k))
    wharmlist.append(sum)

whp=list_plot(zip([2..n],wharmlist),plotjoined=true,color='black')

show(pp+whp)

wharmlist[900]

2.7123610790269677

sumlist[prime_pi(900)]

2.1844657290250784

prime_range(900,1000)

[907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]

for k in [900..950]:
    print wharmlist[k]-sumlist[prime_pi(k)]

0.527895350002 0.528058068905 0.528220581381 0.528382887923 0.528544989022 0.528706885171 0.528868576856 0.527932369726 0.528093653946 0.52825473516 0.528415613849 0.527488151212 0.527648626292 0.527808900283 0.527968973661 0.528128846899 0.52828852047 0.528447994844 0.52860727049 0.527689921609 0.527848801198 0.528007483455 0.528165968843 0.528324257823 0.528482350853 0.528640248392 0.528797950895 0.528955458816 0.52911277261 0.528202656868 0.528359583757 0.528516317869 0.528672859648 0.528829209541 0.528985367991 0.52914133544 0.52929711233 0.528389999843 0.52854539693 0.52870060477 0.528855623798 0.52795448824 0.528109130944 0.528263586132 0.528417854234 0.528571935676 0.528725830885 0.527830222343 0.527983746359 0.528137085412 0.528290239922

mangoldt(7)

Error in lines 1-1 Traceback (most recent call last): File "/projects/cec84faa-0c9f-4d53-a7fe-4962a22dc313/.sagemathcloud/sage_server.py", line 879, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'mangoldt' is not defined

von_mangoldt(7)

Error in lines 1-1 Traceback (most recent call last): File "/projects/cec84faa-0c9f-4d53-a7fe-4962a22dc313/.sagemathcloud/sage_server.py", line 879, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'von_mangoldt' is not defined

mangoldt_lambda(7)

Error in lines 1-1 Traceback (most recent call last): File "/projects/cec84faa-0c9f-4d53-a7fe-4962a22dc313/.sagemathcloud/sage_server.py", line 879, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'mangoldt_lambda' is not defined

mod(5^(-1),16)

13

n=62884891
e=7937
d=mod(e^(-1),euler_phi(n))
d

14859809

n

391

legendre_symbol(31,73)

-1

9973.is_prime()

True

for k in [33..36]:
    print k, legendre_symbol(k,73)

33 -1 34 -1 35 1 36 1

a=8
b=53
legendre_symbol(a,b)

-1

a=54
b=7
legendre_symbol(a,b)

-1

a=15
b=31
legendre_symbol(a,b)

-1

p=17
sum=0
h=(p-1)/2
for k in [1..(h-1)]:
    sum += legendre_symbol(k*(k+1),p)
sum

-1

p=29
sum=0
h=(p-1)/2
for k in [1..h]:
    print k, legendre_symbol(k*(k+1),p), legendre_symbol(k,p)

1 -1 1 2 1 -1 3 -1 -1 4 1 1 5 1 1 6 1 1 7 -1 1 8 -1 -1 9 -1 1 10 1 -1 11 1 -1 12 -1 -1 13 -1 1 14 1 -1

p=11
sum=0
prod=1
h=(p-1)
for k in [1..h]:
    prod *= legendre_symbol(k,p)
    print k, legendre_symbol(k*(k+1),p), prod,legendre_symbol(k,p)

1 -1 1 1 2 -1 -1 -1 3 1 -1 1 4 1 -1 1 5 -1 -1 1 6 1 1 -1 7 1 -1 -1 8 -1 1 -1 9 -1 1 1 10 0 -1 -1

p=11
for k in [1..(p-1)]:
    i2=mod(2^(-1),p)
    ki=mod(k^(-1),p)
    x = i2*(k+ki)
    y = i2*(k-ki)
    print k, ki, x,y, x^2, y^2

1 1 1 0 1 0 2 6 4 9 5 4 3 4 9 5 4 3 4 3 9 6 4 3 5 9 7 9 5 4 6 2 4 2 5 4 7 8 2 5 4 3 8 7 2 6 4 3 9 5 7 2 5 4 10 10 10 0 1 0

p=11
for k in [1..(p-1)]:
    print k, legendre_symbol(k,p)

1 1 2 -1 3 1 4 1 5 1 6 -1 7 -1 8 -1 9 1 10 -1

p=19
h=int((p+1)/2)
#print h
for k in [1..h]:
    i2=mod(2^(-1),p)
    ki=mod(k^(-1),p)
    x = i2*(k+ki)
    y = i2*(k-ki)
    print k,legendre_symbol(k,p),ki, x,y, x^2, y^2

1 1 1 1 0 1 0 2 -1 10 6 15 17 16 3 -1 13 8 14 7 6 4 1 5 14 9 6 5 5 1 4 14 10 6 5 6 1 16 11 14 7 6 7 1 11 9 17 5 4 8 -1 12 10 17 5 4 9 1 17 13 15 17 16 10 -1 2 6 4 17 16

p=17
m=3
headerRow=['k']+['QR?']+['1/k']+['3/k']+['sum']+['diff']
data=[headerRow]
for r in [1..p-1]:
    kminv=m*mod(r^(-1),p)
    data.append([r]+[legendre_symbol(r,p)]+[mod(r^(-1),p)]+[kminv]+[])
table(data,header_row=True, header_column=True)

k | QR? 1/k 3/k +----+-----+-----+-----+ 1 | 1 1 3 2 | 1 9 10 3 | -1 6 1 4 | 1 13 5 5 | -1 7 4 6 | -1 3 9 7 | -1 5 15 8 | 1 15 11 9 | 1 2 6 10 | -1 12 2 11 | -1 14 8 12 | -1 10 13 13 | 1 4 12 14 | -1 11 16 15 | 1 8 7 16 | 1 16 14

5

5

%var x, y, z, w
p=x^5+y^5+z^5+w^5+x^3*y*w+y^3*z*x+z^3*w*y+w^3*x*z+y^3*x*w+x^3*y*z+x^3+y^3+z^3+z^3*x*y+w^3*y*z+x^3*z*w+y^3*z*w+z^3*x*w+w^3+w^3*x*y

p

w^5 + x^5 + w^3*x*y + w*x^3*y + w*x*y^3 + y^5 + w^3*x*z + w*x^3*z + w^3*y*z + x^3*y*z + w*y^3*z + x*y^3*z + w*x*z^3 + w*y*z^3 + x*y*z^3 + z^5 + w^3 + x^3 + y^3 + z^3

s1=x+y+z+w
s2=x*y+x*z+x*w+y*z+y*w+z*w
s3=x*y*z+x*y*w+y*z*w+x*z*w
s4=x*y*z*w

q=s1^3*s2^4*s3^3

q

(w*x + w*y + x*y + w*z + x*z + y*z)^4*(w + x + y + z)^3*w^3*x^3*y^3*z^3

expand(p-s1^5)

-5*w^4*x - 10*w^3*x^2 - 10*w^2*x^3 - 5*w*x^4 - 5*w^4*y - 19*w^3*x*y - 30*w^2*x^2*y - 19*w*x^3*y - 5*x^4*y - 10*w^3*y^2 - 30*w^2*x*y^2 - 30*w*x^2*y^2 - 10*x^3*y^2 - 10*w^2*y^3 - 19*w*x*y^3 - 10*x^2*y^3 - 5*w*y^4 - 5*x*y^4 - 5*w^4*z - 19*w^3*x*z - 30*w^2*x^2*z - 19*w*x^3*z - 5*x^4*z - 19*w^3*y*z - 60*w^2*x*y*z - 60*w*x^2*y*z - 19*x^3*y*z - 30*w^2*y^2*z - 60*w*x*y^2*z - 30*x^2*y^2*z - 19*w*y^3*z - 19*x*y^3*z - 5*y^4*z - 10*w^3*z^2 - 30*w^2*x*z^2 - 30*w*x^2*z^2 - 10*x^3*z^2 - 30*w^2*y*z^2 - 60*w*x*y*z^2 - 30*x^2*y*z^2 - 30*w*y^2*z^2 - 30*x*y^2*z^2 - 10*y^3*z^2 - 10*w^2*z^3 - 19*w*x*z^3 - 10*x^2*z^3 - 19*w*y*z^3 - 19*x*y*z^3 - 10*y^2*z^3 - 5*w*z^4 - 5*x*z^4 - 5*y*z^4 + w^3 + x^3 + y^3 + z^3

expand(p-s1^5+5*s1^3*s2)

5*w^3*x^2 + 5*w^2*x^3 + 16*w^3*x*y + 30*w^2*x^2*y + 16*w*x^3*y + 5*w^3*y^2 + 30*w^2*x*y^2 + 30*w*x^2*y^2 + 5*x^3*y^2 + 5*w^2*y^3 + 16*w*x*y^3 + 5*x^2*y^3 + 16*w^3*x*z + 30*w^2*x^2*z + 16*w*x^3*z + 16*w^3*y*z + 75*w^2*x*y*z + 75*w*x^2*y*z + 16*x^3*y*z + 30*w^2*y^2*z + 75*w*x*y^2*z + 30*x^2*y^2*z + 16*w*y^3*z + 16*x*y^3*z + 5*w^3*z^2 + 30*w^2*x*z^2 + 30*w*x^2*z^2 + 5*x^3*z^2 + 30*w^2*y*z^2 + 75*w*x*y*z^2 + 30*x^2*y*z^2 + 30*w*y^2*z^2 + 30*x*y^2*z^2 + 5*y^3*z^2 + 5*w^2*z^3 + 16*w*x*z^3 + 5*x^2*z^3 + 16*w*y*z^3 + 16*x*y*z^3 + 5*y^2*z^3 + w^3 + x^3 + y^3 + z^3

expand(p-s1^5+5*s1^3*s2-5*s1*s2^2)

6*w^3*x*y + 5*w^2*x^2*y + 6*w*x^3*y + 5*w^2*x*y^2 + 5*w*x^2*y^2 + 6*w*x*y^3 + 6*w^3*x*z + 5*w^2*x^2*z + 6*w*x^3*z + 6*w^3*y*z + 15*w^2*x*y*z + 15*w*x^2*y*z + 6*x^3*y*z + 5*w^2*y^2*z + 15*w*x*y^2*z + 5*x^2*y^2*z + 6*w*y^3*z + 6*x*y^3*z + 5*w^2*x*z^2 + 5*w*x^2*z^2 + 5*w^2*y*z^2 + 15*w*x*y*z^2 + 5*x^2*y*z^2 + 5*w*y^2*z^2 + 5*x*y^2*z^2 + 6*w*x*z^3 + 6*w*y*z^3 + 6*x*y*z^3 + w^3 + x^3 + y^3 + z^3

expand(p-s1^5+5*s1^3*s2-5*s1*s2^2-6*s1^2*s3)

-7*w^2*x^2*y - 7*w^2*x*y^2 - 7*w*x^2*y^2 - 7*w^2*x^2*z - 27*w^2*x*y*z - 27*w*x^2*y*z - 7*w^2*y^2*z - 27*w*x*y^2*z - 7*x^2*y^2*z - 7*w^2*x*z^2 - 7*w*x^2*z^2 - 7*w^2*y*z^2 - 27*w*x*y*z^2 - 7*x^2*y*z^2 - 7*w*y^2*z^2 - 7*x*y^2*z^2 + w^3 + x^3 + y^3 + z^3

expand(p-s1^5+5*s1^3*s2-5*s1*s2^2-6*s1^2*s3+7*s2*s3)

-6*w^2*x*y*z - 6*w*x^2*y*z - 6*w*x*y^2*z - 6*w*x*y*z^2 + w^3 + x^3 + y^3 + z^3

expand(p-s1^5+5*s1^3*s2-5*s1*s2^2-6*s1^2*s3+7*s2*s3+6*s1*s4)

w^3 + x^3 + y^3 + z^3

expand(p-s1^5+5*s1^3*s2-5*s1*s2^2-6*s1^2*s3+7*s2*s3+6*s1*s4-s1^3+3*s1*s2-3*s3)

0

expand(s1^2*s4)

w^3*x*y*z + 2*w^2*x^2*y*z + w*x^3*y*z + 2*w^2*x*y^2*z + 2*w*x^2*y^2*z + w*x*y^3*z + 2*w^2*x*y*z^2 + 2*w*x^2*y*z^2 + 2*w*x*y^2*z^2 + w*x*y*z^3

%var x, y,z
p=x^3+y^3+z^3
e = SymmetricFunctions(QQ).elementary()

#f = e.from_polynomial(p)
#e(f)

e

Symmetric Functions over Rational Field in the elementary basis

e[1]

e[1]

e.from_polynomial?

File: /projects/sage/sage-6.7/local/lib/python2.7/site-packages/sage/combinat/sf/sfa.py
Signature : e.from_polynomial(self, poly, check=True)
Docstring :
Convert polynomial to a symmetric function in the monomial basis
and then to the basis "self".

INPUT:

* "poly" -- a symmetric polynomial

* "check" -- (default: "True") boolean, specifies whether the
  computation checks that the polynomial is indeed symmetric

EXAMPLES:

   sage: Sym = SymmetricFunctions(QQ)
   sage: h = Sym.homogeneous()
   sage: f = (h([]) + h([2,1]) + h([3])).expand(3)
   sage: h.from_polynomial(f)
   h[] + h[2, 1] + h[3]
   sage: s = Sym.s()
   sage: g = (s([]) + s([2,1])).expand(3); g
   x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + 1
   sage: s.from_polynomial(g)
   s[] + s[2, 1]

a=sqrt(3); b=sqrt(5); c= a*b

c

sqrt(5)*sqrt(3)

d=(1-a)*(1-b)*(1-c)*(a-b)*(a-c)*(b-c)

d^2

(sqrt(5)*sqrt(3) - sqrt(5))^2*(sqrt(5)*sqrt(3) - sqrt(3))^2*(sqrt(5)*sqrt(3) - 1)^2*(sqrt(5) - sqrt(3))^2*(sqrt(5) - 1)^2*(sqrt(3) - 1)^2

n(d^2)

21.0854297480639

expand(d^2)

-46080*sqrt(5)*sqrt(3) + 40320*sqrt(5) - 107520*sqrt(3) + 274560

expand(d)

-56*sqrt(5)*sqrt(3) + 192*sqrt(5) - 120*sqrt(3)

[ 1 pi 3] [ e 5 6]

@interact(layout=[['A','Myvect']])
def _(A = matrix([[1,-1],[0,1]]),Myvect=matrix([[0,-1]])):    
    V = vector([1,3])
    W = A*V
    V1 = vector([-1,2])
    W1 = A*V1
    L = line([V,V1],linestyle='--')
    LW = line([W,W1],linestyle='--')
    Myvect = vector(Myvect[0])
    Wyvect = A*Myvect
    show(graphics_array([[plot(V)+plot(V1,color='red')+plot(Myvect,color='green')+plot(L),plot(W)+plot(W1,color='red')+plot(Wyvect,color='green')+plot(LW)]]),aspect_ratio=1) 

Interact: please open in CoCalc

var('x1,x2')
@interact
def _(M = matrix(RR,[[1,5],[-2,-7]]), V = matrix(RR,[[7,-5]]),range=(5,(1,100)),show_sol=False):
    a, b, c, d = M[0,0],M[0,1],M[1,0],M[1,1]
    e, f = V[0,0],V[0,1]
    E1 = a*x1+b*x2==e
    E2 = c*x1+d*x2==f
    eq1 = solve(E1,x2)[0].rhs()
    eq2 = solve(E2,x2)[0].rhs()
    html("Solving:")
    html('Eq 1: $'+latex(E1)+'$')
    html('Eq 2: $'+latex(E2)+'$')
    P = plot(eq1,(x1,-range,range))+plot(eq2,(x1,-range,range),color='red')
    show(P)
    if show_sol:
        try:
            html(solve([E1,E2],[x1,x2])[0])z
        except:
            html('No solution?') 

(x1, x2)

Interact: please open in CoCalc

z=cos(2*pi/5)+i*sin(2*pi/5)

z

1/4*sqrt(5) + 1/4*I*sqrt(2*sqrt(5) + 10) - 1/4

z^5

1/1024*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^5

n(z^5)

1.00000000000000 - 5.55111512312578e-17*I

n(z)

0.309016994374947 + 0.951056516295154*I

def norm(coeffs):
    #coeffs is a list of 5 coefficients a_k*z^k
    z=cos(2*pi/5)+i*sin(2*pi/5)
    prod=1
    for n in [1..4]:
        s=0
        for j in range(5):
            s += coeffs[j]*z^(n*j)
        prod *= s
    return prod

n(norm([0, .5, .0, .5,.4]))

0.0341000000000002 - 3.81639164714898e-17*I

n(norm([2,2,2,2,2]))

3.89424627934904e-60 - 2.02734169599070e-60*I

n=50
s=0
for k in [0..n]:
    s += (-1)^k*(k+1)/factorial(k)
float(s)

3.287949416633158e-65

def d(a,b):
    return abs(a-b)

def delist(myList):
    output = []
    leng = len(myList)
    for c in range(leng):
        output.append(d(myList[c],myList[(c+1)%leng]))
    return output

def size(myList):
    output = 0
    for a in myList:
        output += a*a
    return output
    
    
mylist

Error in lines 14-14 Traceback (most recent call last): File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'mylist' is not defined

delist(mylist)

[5, 1, 2, 4, 6]

size(mylist)

180

len(mylist)

3

mylist[-1]

3

mylist[1:2]

[4]

mylist.mylist

Error in lines 1-1 Traceback (most recent call last): File "/projects/cec84faa-0c9f-4d53-a7fe-4962a22dc313/.sagemathcloud/sage_server.py", line 881, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> AttributeError: 'list' object has no attribute 'mylist'

n=20
trib=[0,1,1]
for k in range(n):
    trib.append(trib[-1]+trib[-2]+trib[-3])

trib

[0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317]

mylist = [23, 11, 15, 10]
while size(mylist)!= 0:
    print mylist
    mylist = delist(mylist)
    

[23, 11, 15, 10] [12, 4, 5, 13] [8, 1, 8, 1] [7, 7, 7, 7]

mylist = [7, 13, 24, 44]
rows =1
while (size(mylist)!= 0)&(rows<40):
    print mylist
    rows += 1
    mylist = delist(mylist)

[7, 13, 24, 44] [6, 11, 20, 37] [5, 9, 17, 31] [4, 8, 14, 26] [4, 6, 12, 22] [2, 6, 10, 18] [4, 4, 8, 16] [0, 4, 8, 12] [4, 4, 4, 12] [0, 0, 8, 8] [0, 8, 0, 8] [8, 8, 8, 8]

mylist = [1, 2, 4, 8]
rows =1
while (size(mylist)!= 0)&(rows<40):
    print mylist
    rows += 1
    mylist = delist(mylist)

[1, 2, 4, 8] [1, 2, 4, 7] [1, 2, 3, 6] [1, 1, 3, 5] [0, 2, 2, 4] [2, 0, 2, 4] [2, 2, 2, 2]

mylist = [1, 4, 9, 16]
rows =1
while (size(mylist)!= 0)&(rows<40):
    print mylist
    rows += 1
    mylist = delist(mylist)

[1, 4, 9, 16] [3, 5, 7, 15] [2, 2, 8, 12] [0, 6, 4, 10] [6, 2, 6, 10] [4, 4, 4, 4]

mylist = [16, 25, 36, 49]
rows =1
while (size(mylist)!= 0)&(rows<40):
    print mylist
    rows += 1
    mylist = delist(mylist)

[16, 25, 36, 49] [9, 11, 13, 33] [2, 2, 20, 24] [0, 18, 4, 22] [18, 14, 18, 22] [4, 4, 4, 4]

mylist = [81, 149, 274, 504]
rows =1
while (size(mylist)!= 0)&(rows<40):
    print mylist
    rows += 1
    mylist = delist(mylist)

[81, 149, 274, 504] [68, 125, 230, 423] [57, 105, 193, 355] [48, 88, 162, 298] [40, 74, 136, 250] [34, 62, 114, 210] [28, 52, 96, 176] [24, 44, 80, 148] [20, 36, 68, 124] [16, 32, 56, 104] [16, 24, 48, 88] [8, 24, 40, 72] [16, 16, 32, 64] [0, 16, 32, 48] [16, 16, 16, 48] [0, 0, 32, 32] [0, 32, 0, 32] [32, 32, 32, 32]

Interact: please open in CoCalc

6*7

42

67^67

5

5

67**67

222337020242360576812569226538683753874082408437758291741262115823894811650848346334502642370010973465496690788650052277723

4.5 %2

0.500000000000000

5.5%2

-0.500000000000000

mod(8,5)

3

mod(5,2)

1

abs(-4)

4

def h(x):
    r=x-2*floor(x/2)
    return abs(r)

h(17)

1

h(-3)

1

plot(h(x),x,-5,5)

3.

3.is_even()

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/structure/element.pyx", line 418, in sage.structure.element.Element.__getattr__ (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/element.c:4670) return getattr_from_other_class(self, P._abstract_element_class, name) File "sage/structure/misc.pyx", line 259, in sage.structure.misc.getattr_from_other_class (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/misc.c:1771) raise dummy_attribute_error AttributeError: 'sage.rings.integer.Integer' object has no attribute 'is_even'

abs

int(9.9)

9

plot(s,.6,.602)

n=100
def h(x):
    x=abs(x)
    if int(x) %2 ==0:
        return x-int(x)
    else:
        return 1-(x-int(x))
    
def s(x):
    v=0
    for i in [0..n]:
        v += h(x*2^i)/2^i
    return v
@interact

def _(x=slider(0,1,default=0.5),maglevel=selector([0..10],nrows=1),mag=checkbox(false,'mag on')):
    if mag:
        d=1/4^maglevel
    else:
        d=x/2
    xmn=x-d
    ymn=s(x)-d
    xmx=x+d
    ymx = ymn+2*d
    p=plot(s,xmin=xmn, ymin=ymn, xmax=xmx, ymax=ymx)
      
        
    
    show(p)

Interact: please open in CoCalc

mlist=[1,2,3,4,5,6,7,8,9]

mlist

[1, 2, 3, 4, 5, 6, 7, 8, 9]

n=200000
myList=[1..n]
sums=[0]
for i in [1..10050]:
    s=sum(myList[:3])
    myList=myList[3:]
    myList.remove(s)
    sums.append(s)
    

[sums[k]+2*k for k in [1..10]]

[8, 20, 33, 44, 56, 69, 80, 91, 105, 116]

lasts=[sums[k]%10000 for k in [0..10000]]

lasts.index(2015)

1202

sums[1202]
sum([sums[k] for k in [1..29]])

12015 4241

lasts.index(75)

8

1202-8

1194

(2015-75)/30
sum([sums[k] for k in [1..u]])+sum([sums[k] for k in [1..3*u-1]])
(10*u-3)*(5*u-1)

194/3

Error in lines 2-2 Traceback (most recent call last): File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'u' is not defined

mlist[3:]

[4, 6, 7, 8, 9]

[sums[k] for k in [1..22]]

sums[8]

75

182+1138

1320

4

4

4

47*48

2256

2256/2

1128

plot((x-2)^2*(x+4)^3/((x-1)^6*(x+2)^3), (x,-6,-2.5))

p=15
q=1
n= p*q
wf=float(cos(2*pi/n))+I*float(sin(2*pi/n))
w=cos(2*pi/n)+I*sin(2*pi/n)

pr=1
for a in [1..n]:
    for b in [1..n]:
        pr *= 1.+w^(a*b)
pr

-(3.19291469757126e-13005)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^225 - 1.56764265941035e203)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^210 - 4.45550841564668e189)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^196 - 1.01306532443384e177)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^195 - 1.26633165554230e176)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^182 - 2.30344386280612e164)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^180 - 3.59913103563456e162)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^169 - 4.18993997810706e152)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^168 - 5.23742497263383e151)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^165 - 1.02293456496754e149)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^156 - 7.62145642166990e140)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^154 - 1.19085256588592e139)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^150 - 2.90735489718243e135)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^144 - 1.10906787764833e130)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^143 - 1.38633484706041e129)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^140 - 2.70768524816486e126)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^135 - 8.26319960987811e121)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^132 - 1.61390617380432e119)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^130 - 2.52172839656925e117)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^126 - 6.15656346818664e113)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^121 - 1.87883406621907e109)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^120 - 2.34854258277383e108)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^117 - 4.58699723198014e105)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^112 - 1.39984046386113e101)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^110 - 2.18725072478301e99)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^108 - 3.41757925747346e97)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^105 - 6.67495948725284e94)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^104 - 8.34369935906606e93)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^100 - 2.03703597633449e90)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^99 - 2.54629497041811e89)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^98 - 3.18286871302263e88)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^96 - 4.97323236409787e86)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^91 - 1.51771007205135e82)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^90 - 1.89713759006419e81)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^88 - 2.96427748447529e79)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^84 - 7.23700557733226e75)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^81 - 1.41347765182271e73)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^80 - 1.76684706477838e72)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^78 - 2.76069853871623e70)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^77 - 3.45087317339528e69)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^75 - 5.39198933343013e67)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^72 - 1.05312291668557e65)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^70 - 1.64550455732121e63)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^66 - 4.01734511064748e59)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^65 - 5.02168138830934e58)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^64 - 6.27710173538668e57)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^63 - 7.84637716923335e56)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^60 - 1.53249554086589e54)^6*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^56 - 3.74144419156711e50)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^55 - 4.67680523945889e49)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^54 - 5.84600654932361e48)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^52 - 9.13438523331814e46)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^50 - 1.42724769270596e45)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^49 - 1.78405961588245e44)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^48 - 2.23007451985306e43)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^45 - 4.35561429658801e40)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^44 - 5.44451787073502e39)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^42 - 8.50705917302346e37)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^40 - 1.32922799578492e36)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^39 - 1.66153499473114e35)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^36 - 3.24518553658427e32)^5*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^35 - 4.05648192073033e31)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^33 - 6.33825300114115e29)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^32 - 7.92281625142643e28)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^30 - 1.23794003928538e27)^6*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^28 - 1.93428131138341e25)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^27 - 2.41785163922926e24)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^26 - 3.02231454903657e23)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^25 - 3.77789318629572e22)*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^24 - 4.72236648286965e21)^6*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^22 - 7.37869762948382e19)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^21 - 9.22337203685478e18)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^20 - 1.15292150460685e18)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^18 - 1.80143985094820e16)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^16 - 2.81474976710656e14)^3*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^15 - 3.51843720888320e13)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^14 - 4.39804651110400e12)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^13 - 5.49755813888000e11)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^12 - 6.87194767360000e10)^6*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^11 - 8.58993459200000e9)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^10 - 1.07374182400000e9)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^9 - 1.34217728000000e8)^3*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^8 - 1.67772160000000e7)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^7 - 2.09715200000000e6)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^6 - 262144.000000000)^4*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^5 - 32768.0000000000)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^4 - 4096.00000000000)^3*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^3 - 512.000000000000)^2*(-1.00000000000000*(sqrt(5) + 2*sqrt(-3/2*sqrt(5) + 15/2) + 8*I*sin(2/15*pi) + 1)^2 - 64.0000000000000)^2*(-1.00000000000000*sqrt(5) - 2.00000000000000*sqrt(-3/2*sqrt(5) + 15/2) - 8.00000000000000*I*sin(2/15*pi) - 9.00000000000000)

real(pr)

/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/functions/other.py:2108: RuntimeWarning: tp_compare didn't return -1 or -2 for exception return GinacFunction.__call__(self, x, **kwargs) Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/functions/other.py", line 2108, in __call__ return GinacFunction.__call__(self, x, **kwargs) File "sage/symbolic/function.pyx", line 847, in sage.symbolic.function.GinacFunction.__call__ (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/function.cpp:9814) res = super(GinacFunction, self).__call__(*args, **kwds) File "sage/symbolic/function.pyx", line 985, in sage.symbolic.function.BuiltinFunction.__call__ (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/function.cpp:11182) res = method() File "sage/symbolic/expression.pyx", line 6859, in sage.symbolic.expression.Expression.real_part (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/expression.cpp:38408) g_hold_wrapper(g_real_part, self._gobj, hold)) File "sage/rings/number_field/number_field_element.pyx", line 1763, in sage.rings.number_field.number_field_element.NumberFieldElement.__pow__ (/projects/sage/sage-6.9/src/build/cythonized/sage/rings/number_field/number_field_element.cpp:18215) return generic_power_c(base, exp, None) File "sage/structure/element.pyx", line 3637, in sage.structure.element.generic_power_c (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/element.c:29598) elif n < 0: File "sage/symbolic/pynac.pyx", line 1014, in sage.symbolic.pynac.py_is_equal (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/pynac.cpp:10389) return bool(x==y) File "sage/symbolic/pynac.pyx", line 1014, in sage.symbolic.pynac.py_is_equal (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/pynac.cpp:10389) return bool(x==y) File "sage/symbolic/pynac.pyx", line 1014, in sage.symbolic.pynac.py_is_equal (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/pynac.cpp:10389) return bool(x==y) File "sage/symbolic/pynac.pyx", line 1014, in sage.symbolic.pynac.py_is_equal (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/pynac.cpp:10389) return bool(x==y) File "sage/symbolic/pynac.pyx", line 1014, in sage.symbolic.pynac.py_is_equal (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/pynac.cpp:10389) return bool(x==y) File "sage/symbolic/pynac.pyx", line 1014, in sage.symbolic.pynac.py_is_equal (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/pynac.cpp:10389) return bool(x==y) File "sage/symbolic/pynac.pyx", line 1045, in sage.symbolic.pynac.py_is_integer (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/pynac.cpp:10525) (x in ZZ)) File "sage/structure/parent.pyx", line 1285, in sage.structure.parent.Parent.__contains__ (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/parent.c:10484) if P is self or P == self: File "sage/rings/integer_ring.pyx", line 365, in sage.rings.integer_ring.IntegerRing_class.__richcmp__ (/projects/sage/sage-6.9/src/build/cythonized/sage/rings/integer_ring.c:4212) return (<Parent>left)._richcmp(right, op) File "sage/structure/parent.pyx", line 1502, in sage.structure.parent.Parent._richcmp (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/parent.c:11961) r = left._cmp_(right) File "sage/rings/integer_ring.pyx", line 383, in sage.rings.integer_ring.IntegerRing_class._cmp_ (/projects/sage/sage-6.9/src/build/cythonized/sage/rings/integer_ring.c:4391) return cmp(type(left), type(right)) File "sage/symbolic/pynac.pyx", line 1014, in sage.symbolic.pynac.py_is_equal (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/pynac.cpp:10389) return bool(x==y) File "sage/rings/number_field/number_field_element_quadratic.pyx", line 672, in sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic.__richcmp__ (/projects/sage/sage-6.9/src/build/cythonized/sage/rings/number_field/number_field_element_quadratic.cpp:7860) return (<Element>left)._richcmp(right, op) File "sage/structure/element.pyx", line 1005, in sage.structure.element.Element._richcmp (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/element.c:9888) left, right = coercion_model.canonical_coercion(self, other) File "sage/structure/coerce.pyx", line 1135, in sage.structure.coerce.CoercionModel_cache_maps.canonical_coercion (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/coerce.c:10171) y_elt = (<Map>y_map)._call_(y) File "sage/structure/coerce_maps.pyx", line 104, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/coerce_maps.c:4432) return C._element_constructor(x) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/rings/number_field/number_field.py", line 1382, in _element_constructor_ def _element_constructor_(self, x, check=True): File "sage/ext/interrupt/interrupt.pyx", line 203, in sage.ext.interrupt.interrupt.sage_python_check_interrupt (/projects/sage/sage-6.9/src/build/cythonized/sage/ext/interrupt/interrupt.c:1890) sig_check() File "sage/ext/interrupt/interrupt.pyx", line 88, in sage.ext.interrupt.interrupt.sig_raise_exception (/projects/sage/sage-6.9/src/build/cythonized/sage/ext/interrupt/interrupt.c:924) raise KeyboardInterrupt KeyboardInterrupt

n(w^15)

0.999999999999999 - 1.38777878078145e-16*I

myl=[1,2,3,4]; myl[-2]

3

a=[1,2]
for n in range(2500):
    a.append(4*a[-1]-a[-2])

a[3]   

26

a[4]

97

a[5]

362

[a[n]%97 for n in [2011..2030]]

[71, 0, 26, 7, 2, 1, 2, 7, 26, 0, 71, 90, 95, 96, 95, 90, 71, 0, 26, 7]

100^2-99^2-98^2+97^2

4

len([1,2,3])

3

mylist1=[1,2,3]
mylist2=[4,5,6]

mylist1+mylist2

[1, 2, 3, 4, 5, 6]

mylist1[:-1]

[1, 2]

def outcomes(mylist):
    lth = len(mylist)
    if lth ==0:
        return []
    else:
        if lth ==1:
            return [mylist[0],-mylist[0]]
        else:
            
            end = mylist[-1]
            first = mylist[:-1]
            temp=outcomes(first)
            outlist = []
            #print outlist
            for v in temp:
                #print v
                outlist.append(v+end)
                outlist.append(v-end)
                #print outlist
            return outlist
 

c=[a%48 for a in outcomes([1,8,27,64,125,216])]

congs = []
m=7
cubes =[]
for k in [1..m]:
    cubes.append(k^3)
    for v in outcomes(cubes):
        congs.append(v%48)
union(congs)

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]

18

len(union(congs))

48

for v in []:
    print 'l'

-(1)^3+2^3+3^3-4^3+5^3-6^3-7^3+8^3

48

s=0
coeffs=[-1,1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1]
total=0
for k in range(16):
    total += coeffs[k]*(s+k)^4
total
    

-32

folds= [1]
f(x) = 1-x


def backflip(myList):
    temp = copy(myList)
    temp.reverse()
    return map(f,temp)

for level in range(12):
    previous=folds
    flipped = backflip(previous)
    
    folds=previous + [1] + flipped
    

len(folds)

8191

for k in range(13):
    print folds[2^k-1]

1 1 1 1 1 1 1 1 1 1 1 1 1

folds[2015]

mlst=[1,0,0,0,1,0]
backflip(mlst)

[1, 0, 1, 1, 1, 0]

mlst

[0, 1, 0, 0, 0, 1]

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> TypeError: argument 2 to map() must support iteration

for k in range(24):
    print k,  7^k

0 1 1 7 2 49 3 343 4 2401 5 16807 6 117649 7 823543 8 5764801 9 40353607 10 282475249 11 1977326743 12 13841287201 13 96889010407 14 678223072849 15 4747561509943 16 33232930569601 17 232630513987207 18 1628413597910449 19 11398895185373143 20 79792266297612001 21 558545864083284007 22 3909821048582988049 23 27368747340080916343

401^5

10368641602001

var('a')
expand((2000*a+1)^5)

a 32000000000000000*a^5 + 80000000000000*a^4 + 80000000000*a^3 + 40000000*a^2 + 10000*a + 1

521973756982990550719199214832958126198121700634846638398592035971765722435787955577804593232551231283468919250766283527579242260482601749889290826725419773851552215257265229938305244969325778738628201036689246273246528854636455945197843566563980378567456558709819621238303165000193416617274725798102234110658591910052267634510706428344516340271478685916960404145585637532327012905727948440551436513376507268061541499810103240257806551355247649156891452359802300657177149222465555301688499164889785395488845823246095892923714630835045456358397990676159773311725816731099440308591085181247212750938104741858370522286597138253543893150043466011612130002639170565743641621933185982688731336278306978131170896030399206747187146615676312653503152705984657867333249498550102237826331911677823704522930573071088299615343901871997962114518154362971166311352852419293587318821327910579860310770963718031815972854318090457698043744252565058078864349773329729872194802436701457281017132473678339929003139321304381497428883796330250687336130949242590503452360900122616992866873335095837908993163504185517957394228304806789601098035733232025570373145284903532292400612865549240753428531158555951619232035853131999862903452202266305564091809805881653610880442679966531140288555025538691780604446100023519241878814424788467460602292643098399301301597561676194009287243216438841925656212180765720469027007908064218550835559977920186197482618713916221783102641478017314936499238242725967667768668965345306766144131520836400377527346892310613598302099630024539336092586281860743378284816789775372168337490660000081669308109945179976300830377876262965625626831138222824795047561068653471967112062191243091976831915011769601

7^2016

521973756982990550719199214832958126198121700634846638398592035971765722435787955577804593232551231283468919250766283527579242260482601749889290826725419773851552215257265229938305244969325778738628201036689246273246528854636455945197843566563980378567456558709819621238303165000193416617274725798102234110658591910052267634510706428344516340271478685916960404145585637532327012905727948440551436513376507268061541499810103240257806551355247649156891452359802300657177149222465555301688499164889785395488845823246095892923714630835045456358397990676159773311725816731099440308591085181247212750938104741858370522286597138253543893150043466011612130002639170565743641621933185982688731336278306978131170896030399206747187146615676312653503152705984657867333249498550102237826331911677823704522930573071088299615343901871997962114518154362971166311352852419293587318821327910579860310770963718031815972854318090457698043744252565058078864349773329729872194802436701457281017132473678339929003139321304381497428883796330250687336130949242590503452360900122616992866873335095837908993163504185517957394228304806789601098035733232025570373145284903532292400612865549240753428531158555951619232035853131999862903452202266305564091809805881653610880442679966531140288555025538691780604446100023519241878814424788467460602292643098399301301597561676194009287243216438841925656212180765720469027007908064218550835559977920186197482618713916221783102641478017314936499238242725967667768668965345306766144131520836400377527346892310613598302099630024539336092586281860743378284816789775372168337490660000081669308109945179976300830377876262965625626831138222824795047561068653471967112062191243091976831915011769601

factor(14182439040)

2^7 * 3^4 * 5 * 7 * 11^2 * 17 * 19

P=Primes()

[P.unrank(k) for k in [100000..100100]]

[1299721, 1299743, 1299763, 1299791, 1299811, 1299817, 1299821, 1299827, 1299833, 1299841, 1299853, 1299869, 1299877, 1299887, 1299899, 1299917, 1299919, 1299941, 1299953, 1299979, 1299989, 1300021, 1300027, 1300031, 1300051, 1300073, 1300097, 1300111, 1300127, 1300129, 1300133, 1300139, 1300141, 1300147, 1300153, 1300181, 1300193, 1300199, 1300237, 1300253, 1300283, 1300289, 1300297, 1300307, 1300309, 1300319, 1300333, 1300339, 1300367, 1300391, 1300421, 1300423, 1300433, 1300451, 1300457, 1300463, 1300471, 1300477, 1300487, 1300501, 1300511, 1300553, 1300571, 1300573, 1300583, 1300597, 1300609, 1300613, 1300633, 1300639, 1300669, 1300681, 1300709, 1300711, 1300727, 1300751, 1300769, 1300771, 1300781, 1300813, 1300829, 1300837, 1300841, 1300843, 1300907, 1300921, 1300927, 1300931, 1300963, 1300967, 1300979, 1300997, 1301011, 1301017, 1301021, 1301023, 1301033, 1301057, 1301077, 1301081, 1301099]

1301021*1301023

1692658244483

1692658244483%9

8

u=0.5
u

0.500000000000000

sqrt(-u*sqrt(u)+u)/u^2

1.53073372946036

f=(1+sqrt(5))/2
a=f; b= -1/f; c= 1-a; d= 1-b

r

a=2+sqrt(3)
b=2-sqrt(3)
c=-1.
d=-1

print a,b,c,d

1/2*sqrt(5) + 1/2 -2/(sqrt(5) + 1) -1/2*sqrt(5) + 1/2 2/(sqrt(5) + 1) + 1

a+b+c+d

2.00000000000000

float(1/a+1/b+1/c+1/d)

1.9999999999999987

float(1/(1-a)+1/(1-b)+1/(1-c)+1/(1-d))

2.0

float(a)

1.618033988749895

float(b)

-0.6180339887498948

float(c)

-0.6180339887498949

float(d)

1.618033988749895

for k in [1..7]:
    u=5^(2^k)
    print k, (u^2-u)/10^k

1 60 2 3900 3 152587500 4 2328306436523437500 5 5421010862427522170037031173706054687500 6 293873587705571876992184134305561419454666388650920794134435709565877914428710937500 7 8636168555094444625386351862800399571116000364436281385023703470168591803162427057971507474084929456003359594727736233040853153480193027391464966058265417814254760742187500

5^16-10000

152587880625

152587880625^2-152587880625

23283061313476662510000

for k in [1..17]:
    u=5^(2^k)%10^k
    print k, u, u^2

1 5 25 2 25 625 3 625 390625 4 625 390625 5 90625 8212890625 6 890625 793212890625 7 2890625 8355712890625 8 12890625 166168212890625 9 212890625 45322418212890625 10 8212890625 67451572418212890625 11 18212890625 331709384918212890625 12 918212890625 843114912509918212890625 13 9918212890625 98370946943759918212890625 14 59918212890625 3590192236006259918212890625 15 259918212890625 67557477392256259918212890625 16 6259918212890625 39186576032079756259918212890625 17 56259918212890625 3165178397321142256259918212890625

5^16

152587890625

for k in [1..20]:
    print k, 5^k

1 5 2 25 3 125 4 625 5 3125 6 15625 7 78125 8 390625 9 1953125 10 9765625 11 48828125 12 244140625 13 1220703125 14 6103515625 15 30517578125 16 152587890625 17 762939453125 18 3814697265625 19 19073486328125 20 95367431640625

for k in [1..30]:
    u=5^(2^k)%10^k
    print k, u

1 5 2 25 3 625 4 625 5 90625 6 890625 7 2890625 8 12890625 9 212890625 10 8212890625 11 18212890625 12 918212890625 13 9918212890625 14 59918212890625 15 259918212890625 16 6259918212890625 17 56259918212890625 18 256259918212890625 19 2256259918212890625 20 92256259918212890625 21 392256259918212890625 22 7392256259918212890625 23 77392256259918212890625 24 977392256259918212890625 25 9977392256259918212890625 26 19977392256259918212890625 27 619977392256259918212890625 28 6619977392256259918212890625 29 6619977392256259918212890625 30 106619977392256259918212890625

6619977392256259918212890625

5

5

6619977392256259918212890625^2

43824100673983991394155879106619977392256259918212890625

9625^2

92640625

62562

62562

625^2

390625

for k in [1..9]:
    print k, 5^(2^k)

1 25 2 625 3 390625 4 152587890625 5 23283064365386962890625 6 542101086242752217003726400434970855712890625 7 293873587705571876992184134305561419454666389193021880377187926569604314863681793212890625 8 86361685550944446253863518628003995711160003644362813850237034701685918031624270579715075034722882265605472939461496635969950989468319466936530037770580747746862471103668212890625 9 7458340731200206743290965315462933837376471534600406894271518333206278385070118304936174890400427803361511603255836101453412728095225302660486164829592084691481260792318781377495204074266435262941446554365063914765414217260588507120031686823003222742297563699265350215337206058336516628646003612927433551846968657326499008153319891789578832685947418212890625

106619977392256259918212890625^2

11367819579125235975036734004106619977392256259918212890625

625^2

390625

7^899

5535259332367700277978657326589927198133008107122538365354192538956547086422515910639636698316809405197004157588908165185273330066488585796724794051175182565589381565565900357737376274323429225672888525668572160105313219760532202235440837604994309541976321319719709618539837068646339433990033331721080114918303005889346193909637364851356692525890440460794334812083737004704271134457490656828562662159325676765590727941985516775771401780048062630125318459180313888360650125531477863690761630557251636930271428040920539024248699891536258196815401632705539823560804034267139224992033028692517856283003110131582042634439135577934405335297898262493410892234857848337092975960637469951317432992282890881496953958018946412622539699551286744074466684710433403136077143

plot(sin,0,pi)

factor(2016)

2^5 * 3^2 * 7

for k in range(8):
    print k, factor(10^(2^k)+1)

0 11 1 101 2 73 * 137 3 17 * 5882353 4 353 * 449 * 641 * 1409 * 69857 5 19841 * 976193 * 6187457 * 834427406578561 6 1265011073 * 15343168188889137818369 * 515217525265213267447869906815873 7 257 * 15361 * 453377 * 55871187633753621225794775009016131346430842253464047463157158784732544216230781165223702155223678309562822667655169

continued_fraction??

   File: /projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/rings/continued_fraction.py
   Source:
   def continued_fraction(x, value=None):
    r"""
    Return the continued fraction of ``x``.

    INPUT:

        - `x` -- a number or a list of partial quotients (for finite
          development) or two list of partial quotients (preperiod and period
          for ultimately periodic development)

    EXAMPLES:

    A finite continued fraction may be initialized by a number or by its list of
    partial quotients::

        sage: continued_fraction(12/571)
        [0; 47, 1, 1, 2, 2]
        sage: continued_fraction([3,2,1,4])
        [3; 2, 1, 4]

    It can be called with elements defined from symbolic values, in which case
    the partial quotients are evaluated in a lazy way::

        sage: c = continued_fraction(golden_ratio); c
        [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...]
        sage: c.convergent(12)
        377/233
        sage: fibonacci(14)/fibonacci(13)
        377/233

        sage: continued_fraction(pi)
        [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...]
        sage: c = continued_fraction(pi); c
        [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...]
        sage: a = c.convergent(3); a
        355/113
        sage: a.n()
        3.14159292035398
        sage: pi.n()
        3.14159265358979

        sage: continued_fraction(sqrt(2))
        [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...]
        sage: continued_fraction(tan(1))
        [1; 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, ...]
        sage: continued_fraction(tanh(1))
        [0; 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, ...]
        sage: continued_fraction(e)
        [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, ...]

    If you want to play with quadratic numbers (such as ``golden_ratio`` and
    ``sqrt(2)`` above), it is much more convenient to use number fields as
    follows since preperiods and periods are computed::

        sage: K.<sqrt5> = NumberField(x^2-5, embedding=2.23)
        sage: my_golden_ratio = (1 + sqrt5)/2
        sage: cf = continued_fraction((1+sqrt5)/2); cf
        [(1)*]
        sage: cf.convergent(12)
        377/233
        sage: cf.period()
        (1,)
        sage: cf = continued_fraction(2/3+sqrt5/5); cf
        [1; 8, (1, 3, 1, 1, 3, 9)*]
        sage: cf.preperiod()
        (1, 8)
        sage: cf.period()
        (1, 3, 1, 1, 3, 9)

        sage: L.<sqrt2> = NumberField(x^2-2, embedding=1.41)
        sage: cf = continued_fraction(sqrt2); cf
        [1; (2)*]
        sage: cf.period()
        (2,)
        sage: cf = continued_fraction(sqrt2/3); cf
        [0; 2, (8, 4)*]
        sage: cf.period()
        (8, 4)

    It is also possible to go the other way around, build a ultimately periodic
    continued fraction from its preperiod and its period and get its value
    back::

        sage: cf = continued_fraction([(1,1),(2,8)]); cf
        [1; 1, (2, 8)*]
        sage: cf.value()
        2/11*sqrt5 + 14/11

    It is possible to deal with higher degree number fields but in that case the
    continued fraction expansion is known to be aperiodic::

        sage: K.<a> = NumberField(x^3-2, embedding=1.25)
        sage: cf = continued_fraction(a); cf
        [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, ...]

    Note that initial rounding can result in incorrect trailing partial
    quotients::

        sage: continued_fraction(RealField(39)(e))
        [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 2]

    Note the value returned for floating point number is the continued fraction
    associated to the rational number you obtain with a conversion::

        sage: for _ in xrange(10):
        ....:     x = RR.random_element()
        ....:     cff = continued_fraction(x)
        ....:     cfe = QQ(x).continued_fraction()
        ....:     assert cff == cfe, "%s %s %s"%(x,cff,cfe)

    TESTS:

    Fixed :trac:`18901`. For RealLiteral, continued_fraction calls
    continued_fraction_list::

        sage: continued_fraction(1.575709393346379)
        [1; 1, 1, 2, 1, 4, 18, 1, 5, 2, 25037802, 7, 1, 3, 1, 28, 1, 8, 2]
    """

    if isinstance(x, ContinuedFraction_base):
        return x

    try:
        return x.continued_fraction()
    except AttributeError:
        pass

    # input for finite or ultimately periodic partial quotient expansion
    from sage.combinat.words.finite_word import FiniteWord_class
    if isinstance(x, FiniteWord_class):
        x = list(x)

    if isinstance(x, (list, tuple)):
        if len(x) == 2 and isinstance(x[0], (list,tuple)) and isinstance(x[1], (list,tuple)):
            x1 = tuple(Integer(a) for a in x[0])
            x2 = tuple(Integer(a) for a in x[1])
            x1, x2 = check_and_reduce_pair(x1, x2)
        else:
            x1, x2 = check_and_reduce_pair(x)
        return ContinuedFraction_periodic(x1, x2)

    # input for infinite partial quotient expansion
    from sage.misc.lazy_list import lazy_list
    from sage.combinat.words.infinite_word import InfiniteWord_class
    if isinstance(x, (lazy_list, InfiniteWord_class)):
        return ContinuedFraction_infinite(x, value)

    from sage.combinat.words.abstract_word import Word_class
    if isinstance(x, Word_class):
        raise ValueError("word with unknown length can not be converted to continued fractions")

    # input for numbers
    #TODO: the approach used below might be not what the user expects as we
    # have currently in sage (version 6.8)
    #
    #     sage: RR.random_element() in QQ
    #     True
    #
    # But, be careful with real literals
    #
    #     sage: a = 1.575709393346379
    #     sage: a in QQ
    #     False
    from rational_field import QQ
    if x in QQ:
        return QQ(x).continued_fraction()

    is_real = False
    try:
        is_real = x.is_real()
    except AttributeError:
        pass

    from real_mpfi import RealIntervalField, RealIntervalFieldElement
    if is_real is False:
        # we can not rely on the answer of .is_real() for elements of the
        # symbolic ring. The thing below is a dirty temporary hack.
        RIF = RealIntervalField(53)
        try:
            RIF(x)
            is_real = True
        except (AttributeError,ValueError):
            pass

    if is_real is False:
        raise ValueError("the number %s does not seem to be a real number"%x)

    if x.parent().is_exact():
        return ContinuedFraction_real(x)

    # we treat separatly the symbolic ring that holds all constants and
    # which is not exact
    from sage.symbolic.ring import SR
    if x.parent() == SR:
        return ContinuedFraction_real(x)

    return continued_fraction(continued_fraction_list(x))

ec1=plot(sqrt(x^3+17),(x,-17^(1/3),5),aspect_ratio=1)
ec2=plot(-sqrt(x^3+17),(x,-17^(1/3),5),aspect_ratio=1)


pt = point((-1,4))
(ec1+ec2+pt).show()

def newx(u):
    v=sqrt(u^3+17)
    m=3*u^2/(2*v)
    return m^2-2*u

newx(-1)

137/64

def twop(pt):
    return [2*pt[0],2*pt[1]]

twop([1,3])

[2, 6]

[3^k % 13 for k in range(12)]

[1, 3, 9, 1, 3, 9, 1, 3, 9, 1, 3, 9]

l2=log(2.,10)

l2*100000

30102.9995663981

2.^100000/10^30103

0.999002093014384

a =799

a.is_prime()

False

def twice(x):
    return x+x

twice(100)

200

twice('cat')

'catcat'

print 'hello world'

hello world

plot(x * sin(x), (x, -2, 10))

g = graphs.RandomGNM(15, 105)  # 15 vertices and 20 edges
show(g,size=1000)
g.incidence_matrix()

d3-based renderer not yet implemented

15 x 105 sparse matrix over Integer Ring

m=7
headerRow=['x']+range(m)
data=[headerRow]
for r in range(m):
    data.append([r]+[mod(r*c,m) for c in range(m)])
table(data,header_row=True, header_column=True)

x | 0 1 2 3 4 5 6 +---+---+---+---+---+---+---+---+ 0 | 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 2 | 0 2 4 6 1 3 5 3 | 0 3 6 2 5 1 4 4 | 0 4 1 5 2 6 3 5 | 0 5 3 1 6 4 2 6 | 0 6 5 4 3 2 1

x=9

x.coprime_integers(9)

[1, 2, 4, 5, 7, 8]

m=8
cops = m.coprime_integers(m)
headerRow=['x']+cops
data=[headerRow]

for r in cops:
    data.append([r]+[mod(r*c,m) for c in cops])
table(data,header_row=True, header_column=True)

x | 1 3 5 7 +---+---+---+---+---+ 1 | 1 3 5 7 3 | 3 1 7 5 5 | 5 7 1 3 7 | 7 5 3 1

m=11
cops = m.coprime_integers(m)
print '&',
for c in cops:
    print c,'&',
print "lr"
for r in cops:
    print r,'&',
    for c in cops:
        print mod(r*c,m),"&",
    print "lr"
        

& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & lr 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & lr 2 & 2 & 4 & 6 & 8 & 10 & 1 & 3 & 5 & 7 & 9 & lr 3 & 3 & 6 & 9 & 1 & 4 & 7 & 10 & 2 & 5 & 8 & lr 4 & 4 & 8 & 1 & 5 & 9 & 2 & 6 & 10 & 3 & 7 & lr 5 & 5 & 10 & 4 & 9 & 3 & 8 & 2 & 7 & 1 & 6 & lr 6 & 6 & 1 & 7 & 2 & 8 & 3 & 9 & 4 & 10 & 5 & lr 7 & 7 & 3 & 10 & 6 & 2 & 9 & 5 & 1 & 8 & 4 & lr 8 & 8 & 5 & 2 & 10 & 7 & 4 & 1 & 9 & 6 & 3 & lr 9 & 9 & 7 & 5 & 3 & 1 & 10 & 8 & 6 & 4 & 2 & lr 10 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & lr

m=11
k=7
[mod(k^r,m) for r in range(m)]

[1, 7, 5, 2, 3, 10, 4, 6, 9, 8, 1]

mod(7^2016,11)

4

f=sin(tan(x))

f.taylor(x,0,15)

-571772647/100590336000*x^15 - 7000033/566092800*x^13 - 968167/39916800*x^11 - 143/3456*x^9 - 55/1008*x^7 - 1/40*x^5 + 1/6*x^3 + x

g=tan(sin(x))
g.taylor(x,0,15)

986527/37362124800*x^15 + 3027637/6227020800*x^13 + 41897/39916800*x^11 - 73/24192*x^9 - 107/5040*x^7 - 1/40*x^5 + 1/6*x^3 + x

(f-g).taylor(x,0,7)

-1/30*x^7

ff=arcsin(arctan(x))
ff.taylor(x,0,15)

-4043494549/145297152000*x^15 + 1087433/32947200*x^13 - 177761/4435200*x^11 + 18649/362880*x^9 - 341/5040*x^7 + 13/120*x^5 - 1/6*x^3 + x

gg=arctan(arcsin(x))
gg.taylor(x,0,15)

-889424791/435891456000*x^15 + 29518679/2075673600*x^13 - 123379/13305600*x^11 + 12409/362880*x^9 - 173/5040*x^7 + 13/120*x^5 - 1/6*x^3 + x

(ff-gg).taylor(x,0,7)

-1/30*x^7

# collatz (n) equals n/2 if n is even, 3n+1 if n is odd

def collatz(n):
    if n%2 == 0:
        return n/2
    else:
        return 3*n+1
        
# compute the inverse as a list (there may be one or two values)        
def collatzInv(n):
    if n%6==4:
        return [2*n, (n-1)/3]
    else:
        return [2*n]
    
# give a list of successive iterations ending at 1        
def collatzEvol(n):
    next = collatz(n)
    evol=[n,next]
    while next <> 1:
        next = collatz(next)
        evol.append(next)
    return evol

# make a list of inverses up to length "level"
def collatzInvEvol(n,level):
    evol = [n]
    while level>0:
        for a in evol:
            evol = union(evol, collatzInv(a))
        level -= 1
    return evol
    
# compute the length of the iteration
def collatzLength(n):
    return len(collatzEvol(n))

#compute the 'max hailstone height'
def collatzMax(n):
    return max(collatzEvol(n))

heights = [collatzMax(a) for a in [1..1000]]
list_plot(heights,ymax=10000)

lev = 12
vals = collatzInvEvol(1,lev)
#vals=union(vals,collatzEvol(9))
#vals=union(vals, collatzEvol(576))
t=17
for i in [3..t]:
    vals=union(vals, collatzEvol(i))
vals.remove(1)
G=DiGraph({a:[collatz(a)] for a in vals})
G.show(figsize=[10,10],vertex_size=700,layout='tree',tree_root=1,tree_orientation='up')