import math
# Berechnet die Summe aller vielfachen von base bis max
-def summandial(m,base=1):
- m = int(m/base)*base
- return m*(m+base)/(2*base)
+def summandial(m, base=1):
+ m = int(m / base) * base
+ return m * (m + base) / (2 * base)
# Gibt die naechst kleine Fib zu max zurueck
-def fib(m,a=1,b=2):
+def fib(m, a=1, b=2):
while b < m:
- a,b = b,a+b
+ a, b = b, a + b
return a
# Gibt die Liste aller Fib zur naechst kleineren Fib von max zurueck
-def fibV(m,a=[1,2]):
+def fibV(m, a=[1, 2]):
i = 1;
while a[i] < m:
- a.append(a[i] + a[i-1])
- i = i+1
+ a.append(a[i] + a[i - 1])
+ i = i + 1
return a
# Gibt die Summe aller d-fachen Fib zu max zurueck (d=2)
-def fibS(m,d=2):
- a,b = 1,2
+def fibS(m, d=2):
+ a, b = 1, 2
s = 0
while b < m:
- a,b = b,a+b
- if a%d == 0:
- s = s +a
+ a, b = b, a + b
+ if a % d == 0:
+ s = s + a
return s
# Gibt alle Primzahlen bis n zurueck
def primes(n):
if n < 2:
return 0
- p = range(1,n+1,2)
+ p = range(1, n + 1, 2)
q = len(p);
p[0] = 2;
- for k in range(3,int(n**.5+1),2):
- if p[(k-1)/2]:
- for i in range(((k*k-1)/2),q,k):
+ for k in range(3, int(n ** .5 + 1), 2):
+ if p[(k - 1) / 2]:
+ for i in range(((k * k - 1) / 2), q, k):
p[i] = 0
- return filter(lambda x:x>0,p)
+ return filter(lambda x:x > 0, p)
# Gibt alle Primfaktoren von n zurueck
def factor(n):
- if n<4:
+ if n < 4:
return [n]
- p = primes(int(n**.5+1))
+ p = primes(int(n ** .5 + 1))
f = []
- while n>1:
- p = filter(lambda x:n%x==0,p)
- if(p==[]):
+ while n > 1:
+ p = filter(lambda x:n % x == 0, p)
+ if(p == []):
f.append(n)
break
f[len(f):] = p
- n = n/prod(p)
+ n = n / prod(p)
f.sort()
return f
# Gibt alle Primfaktoren von n als vektor mit Vielfachheit zurueck
def factorD(n):
- if n<4:
+ if n < 4:
return {n:1}
- p = primes(int(n**.5+1))
+ p = primes(int(n ** .5 + 1))
f = {}
- while n>1:
- p = filter(lambda x:n%x==0,p)
- if(p==[]):
+ while n > 1:
+ p = filter(lambda x:n % x == 0, p)
+ if(p == []):
f[n] = 1
break
for i in range(len(p)):
if(p[i] in f):
- f[p[i]]=f[p[i]]+1
+ f[p[i]] = f[p[i]] + 1
else:
- f[p[i]]=1
- n = n/prod(p)
+ f[p[i]] = 1
+ n = n / prod(p)
# f.sort()
return f
# Multipliziert alle Werte aus Liste seq auf
def prod(seq):
- return reduce(lambda x,y:x*y, seq, 1)
+ return reduce(lambda x, y:x * y, seq, 1)
# generiert Pyth tripel
-def pythtrip(u,v):
- return [u*u-v*v, 2*u*v, u*u+v*v]
+def pythtrip(u, v):
+ return [u * u - v * v, 2 * u * v, u * u + v * v]
#Binomial Koeffizient
-def binomial(n,k):
- return math.factorial(n)/(math.factorial(k)*math.factorial(n-k))
+def binomial(n, k):
+ return math.factorial(n) / (math.factorial(k) * math.factorial(n - k))
def ebinomial(seq):
vek = [1, 0]
for i in seq:
- for j in range(len(vek)-1,0,-1):
- vek[j] = vek[j]+vek[j-1]*i
- vek[len(vek):]=[0]
+ for j in range(len(vek) - 1, 0, -1):
+ vek[j] = vek[j] + vek[j - 1] * i
+ vek[len(vek):] = [0]
- return vek[:len(vek)-1]
+ return vek[:len(vek) - 1]
#Anzahl der Teiler
def divisors(num):
test = factorD(num)
- return sum(ebinomial(map(lambda x:test[x],test.keys()))[1:])+1
+ return sum(ebinomial(map(lambda x:test[x], test.keys()))[1:]) + 1
#erweiterter Euklid
-def euklid(a,b):
+def euklid(a, b):
if a < b:
tmp = a
a = b
x = [1, 0]
y = [0, 1]
q = a / b
- while a%b:
- [a ,b] = [b,a-q*b]
- x = [x[1],x[0]-q*x[1]]
- y = [y[1],y[0]-q*y[1]]
- q = a/b
- return [b,x[1],y[1]]
+ while a % b:
+ [a , b] = [b, a - q * b]
+ x = [x[1], x[0] - q * x[1]]
+ y = [y[1], y[0] - q * y[1]]
+ q = a / b
+ return [b, x[1], y[1]]
#erweiterter Euklid mit TeX Ausgabe
-def euklid_tex(a,b):
+def euklid_tex(a, b):
if a < b:
tmp = a
a = b
q = int(a / b)
print "\\begin{array}{ccccccc}"
print " r_{i-2} & r_{i-1} & q_i & x_{i-2} & x_{i-1} & y_{i-2} & y_{i-1}\\\\\\hline"
- print " ", a, "&" , b ,"&" , q,"&" ,x[0],"&",x[1] ,"&",y[0] ,"&" ,y[1],"\\\\"
- while a%b:
- [a ,b] = [b,a-q*b]
- x = [x[1],x[0]-q*x[1]]
- y = [y[1],y[0]-q*y[1]]
- q = int(a/b)
- if a%b:
- print " ", a, "&" , b ,"&" , q,"&" ,x[0],"&",x[1] ,"&",y[0] ,"&" ,y[1],"\\\\"
+ print " ", a, "&" , b , "&" , q, "&" , x[0], "&", x[1] , "&", y[0] , "&" , y[1], "\\\\"
+ while a % b:
+ [a , b] = [b, a - q * b]
+ x = [x[1], x[0] - q * x[1]]
+ y = [y[1], y[0] - q * y[1]]
+ q = int(a / b)
+ if a % b:
+ print " ", a, "&" , b , "&" , q, "&" , x[0], "&", x[1] , "&", y[0] , "&" , y[1], "\\\\"
else:
print " \\cline{2-2}\\cline{5-5}\\cline{7-7}"
print " \multicolumn{1}{l|}{", a, "}& \multicolumn{1}{l|}{" , b ,
- print "}&&\multicolumn{1}{l|}{" ,x[0],"}&\multicolumn{1}{l|}{",x[1] ,
- print "}&\multicolumn{1}{l|}{",y[0] ,"}&\multicolumn{1}{l|}{" ,y[1], "}\\\\"
+ print "}&&\multicolumn{1}{l|}{" , x[0], "}&\multicolumn{1}{l|}{", x[1] ,
+ print "}&\multicolumn{1}{l|}{", y[0] , "}&\multicolumn{1}{l|}{" , y[1], "}\\\\"
print " \\cline{2-2}\\cline{5-5}\\cline{7-7}"
print "\\end{array}",
- return [b,x[1],y[1]]
+ return [b, x[1], y[1]]
-# groesster gemeinsamer Teiler
-def gcd(a,b):
- return euklid(a,b)[0]
+# groesster gemeinsamer Teiler
+def gcd(a, b):
+ return euklid(a, b)[0]
# kleinstes gemeinsames Vielfaches
-def lcm(a,b):
- return abs(a*b) / gcd(a,b)
+def lcm(a, b):
+ return abs(a * b) / gcd(a, b)
#eulersche phi-Funktion
def phi(n):
- return n*reduce(lambda x,p:x*(1-1./p), factorD(n).keys(), 1)
+ return n * reduce(lambda x, p:x * (1 - 1. / p), factorD(n).keys(), 1)
-def ordm(a,m):
+def ordm(a, m):
e = 1
ae = a
- while ae%m!=1:
- ae=ae*a
- e = e+1
+ while ae % m != 1:
+ ae = ae * a
+ e = e + 1
return e
# Legendre Symbol
-def legendre(z,n):
- stack = [[z,n]]
+def legendre(z, n):
+ stack = [[z, n]]
d = +1
while len(stack):
- [z,n] = stack.pop(0)
- print z,"/",n
- if(z>n):
- stack.append([z%n, n])
+ [z, n] = stack.pop(0)
+# print z,"/",n
+ if(z > n):
+ stack.append([z % n, n])
+ continue
+
+ if(z == n - 1):
+ if(n % 4 == 3):
+ d = d * (-1)
continue
fac = factorD(z).keys()
- if(len(fac)>1):
- stack[len(stack):] = map(lambda x:[x,n],fac)
+ if(len(fac) > 1):
+ stack[len(stack):] = map(lambda x:[x, n], fac)
continue
- if(z==2):
- if(n%8==3 or n%8==5):
- d = d*(-1)
+ if(z == 2):
+ if(n % 8 == 3 or n % 8 == 5):
+ d = d * (-1)
continue
- if(z==1):
+ if(z == 1):
continue
- stack.append([n,z])
- if (n%4==3 and z%4==3):
- d = d* (-1)
+ stack.append([n, z])
+ if (n % 4 == 3 and z % 4 == 3):
+ d = d * (-1)
return d
#Legendre Symbol mit TeX Ausgabe
-def legendre_tex(z,n):
- stack = [[z,n],[0,0]]
- l=-1
+def legendre_tex(z, n):
+ stack = [[z, n], [0, 0]]
+ l = -1
d = +1
print "\\begin{align}"
- while len(stack)>1:
- [z,n] = stack.pop(0)
+ while len(stack) > 1:
+ [z, n] = stack.pop(0)
- if([z,n]==[0,0]):
- stack.append([0,0])
- l = l+1
- if(l==0):
+ if([z, n] == [0, 0]):
+ stack.append([0, 0])
+ l = l + 1
+ if(l == 0):
print "\n&=",
- elif(l>3):
+ elif(l > 3):
print "\\\\ \n&=",
l = 1
else:
print "\n=",
- if(d<0):
+ if(d < 0):
print "-",
continue
- print "\\legend[L]{",z,"}{",n,"}",
+ print "\\legend[L]{", z, "}{", n, "}",
- if(z>n):
- stack.append([z%n, n])
+ if(z > n):
+ stack.append([z % n, n])
+ continue
+
+ if(z == n - 1):
+ if(n % 4 == 3):
+ d = d * (-1)
continue
fac = factorD(z).keys()
- if(len(fac)>1):
- stack[len(stack):] = map(lambda x:[x,n],fac)
+ if(len(fac) > 1):
+ stack[len(stack):] = map(lambda x:[x, n], fac)
continue
- if(z==2):
- if(n%8==3 or n%8==5):
- d = d*(-1)
+ if(z == 2):
+ if(n % 8 == 3 or n % 8 == 5):
+ d = d * (-1)
continue
- if(z==1):
+ if(z == 1):
continue
- stack.append([n,z])
- if (n%4==3 and z%4==3):
- d = d* (-1)
+ stack.append([n, z])
+ if (n % 4 == 3 and z % 4 == 3):
+ d = d * (-1)
+
+ print
+ print "=", d
+ print "\\end{align}"
+ return d
+
+# Jacobi Symbol
+def jacobi(z, n):
+ d = +1
+ while 1:
+ print z, "/", n
+ if(z > n):
+ z = z % n
+ continue
+
+ if(z == n - 1):
+ if(n % 4 == 3):
+ d = d * (-1)
+ break
+
+ if(z % 4 == 0):
+ z = z / 4
+ continue
+
+ if(z == 2):
+ if(n % 8 == 3 or n % 8 == 5):
+ d = d * (-1)
+ break
+ if(z == 1):
+ break
+
+ [z, n] = [n, z]
+ if (n % 4 == 3 and z % 4 == 3):
+ d = d * (-1)
+
+ return d
+
+#Jacobi Symbol mit TeX Ausgabe
+def jacobi_tex(z, n):
+ l = -2
+ d = +1
+ print "\\begin{align}"
+ while 1:
+
+ l = l + 1
+ if(l == 0):
+ print "\n&=",
+ elif(l > 3):
+ print "\\\\ \n&=",
+ l = 1
+ elif(l>0):
+ print "\n=",
+ if(d < 0):
+ print "-",
+
+ print "\\legend[J]{", z, "}{", n, "}",
+
+
+ if(z > n):
+ z = z % n
+ continue
+
+ if(z == n - 1):
+ if(n % 4 == 3):
+ d = d * (-1)
+ break
+
+ if(z % 4 == 0):
+ z = z / 4
+ continue
+
+ if(z == 2):
+ if(n % 8 == 3 or n % 8 == 5):
+ d = d * (-1)
+ break
+ if(z == 1):
+ break
+
+ [z, n] = [n, z]
+ if (n % 4 == 3 and z % 4 == 3):
+ d = d * (-1)
print
- print "=",d
+ print "=", d
print "\\end{align}"
return d
-legendre_tex(69,97)
\ No newline at end of file
+print jacobi_tex(69, 97)
projects / zahlenTA.git / commitdiff