def convergent(i, n): base = int(sqrt(i)) c = [base] diff = -base numerator = 1 for z in range(1, n): d = i - diff**2 if not gcd(numerator, d) == numerator: raise "ERROR" d /= numerator numeratordiff = -diff x = 0 while abs(numeratordiff-d) <= base: numeratordiff -= d x += 1 c.append(x) diff = numeratordiff numerator = d # compute output a = c.pop() d = 1 while len(c): b = a a = a*c.pop() + d d = b return (a, d)
def Z(k): r = 0 for i in range(10, k+1): if i%10 == 0: r += 1 else: t = i/10**(len(str(i))-1) if gcd(i%10, t) == t: r += 1 return r
def generate_keys(self):
print('Generating keys for RSA')
_bits = 8
p, q = randPrime(_bits), randPrime(_bits)
while (p == q):
p = randPrime(3)
q = randPrime(3)
# print ('p =',p,', q =',q)
n = p * q
phi = (p - 1) * (q - 1)
# an e that makes E and PHI coprime
e = random.randrange(3, phi - 1)
# verify coprime-ness
g = gcd(e, phi)
while g != 1:
e = random.randrange(3, phi - 1)
g = gcd(e, phi)
d = modInv(e, phi)
# self.public_key = (n, e)
# self.private_key = (n, d)
return ((n, e), (n, d)) #public, private
# 30872nd
import tools
def multiplicative_order(a, n):
base = a
for i in range(1, n):
if a % n == 1:
return i
a *= base
if __name__ == "__main__":
max = (0, 0)
for i in range(2, 1000):
if tools.gcd(i, 10) == 1:
x = multiplicative_order(10, i)
if x > max[1]:
max = (x, i)
print max
limit = 1500000 i = 12 result = 0 primitives = [] p = 0 ps = set() ds = set() for m in range(2, 867): for n in range(1, m): if m%2 == n%2: continue if tools.gcd(m, n) == 1: p += 1 x = 2*m*(m+n) if x in ps: ds.add(x) ps.add(x) while i <= limit: j = i/2 if tools.isprime(j): i += 2 continue (f, c) = tools.primefactorization(primes, j, 1000000)
import tools target = 8 target = 1000000 resultn = 1 resultd = target for d in xrange(2, target+1): s = max(1, int(d*resultn*1.0/resultd)) #print s*1.0/d, 3.0/7, s*1.0/d <= resultn*1.0/resultd for n in xrange(s, int(d*3.0/7.0)+1): if tools.gcd(n, d)-1: continue if resultn*1.0/resultd < n*1.0/d and 3.0/7.0 > n*1.0/d: resultn = n resultd = d print resultn, resultd
def sqrt_mod_prime_pow(x, p, n):
if x % p:
return hensel_lift(x, p, n)
else:
res = []
pn = pow(p, n)
y = mod(x, pn)
# Case 1
if y == 0:
m, i = n >> 1, 0
pm = pow(p, m + 1) if (n & 1) == 1 else pow(p, m)
while i < pn:
res.append(i)
i += pm
return res
else:
g = gcd(x, pn)
r = 0
# Extract factors of the prime from the gcd
while g % p == 0:
g /= p
r += 1
if (r & 1) == 1:
return None
m = r >> 1
x1 = x >> r
if p == 2:
# Case 2a
if n - r == 1:
pmn1 = 1 << (n - m + 1)
pm1 = 1 << (m + 1)
k, i = pm1 << 1, pm1 >> 1
while i < pmn1:
j = i
while j < pn:
res.append(j)
j += k
i += pm1
# Case 2b
elif n - r == 2:
res1 = hensel_lift(x1, p, n - r)
if res1 is None:
return None
pnm, s = 1 << (n - m), set([])
for r in res1:
i = 0
while i < pn:
x = (r << m) + i
s.add(x)
i += pnm
res = list(s)
# Case 2c
else:
res1 = hensel_lift(x1, p, n - r)
if res1 is None:
return None
pnm1, s = 1 << (n - m - 1), set([])
for r in res1:
i = 0
while i < pn:
x = mod((r << m) + i, pn)
s.add(x)
i += pnm1
res = list(s)
# case 3
else:
m = r >> 1
x1 = x // pow(p, r)
res1 = hensel_lift(x1, p, n - r)
if res1 is None:
return None
res1 = res1[0]
pm = pow(p, m)
pnr = pow(p, n - r)
pnm = pow(p, n - m)
i = 0
while i < pnm:
x = mod(res1 + i, pn)
res.append(x * pm)
i += pnr
return res
#30872nd
import tools
def multiplicative_order(a, n):
base = a
for i in range(1, n):
if a % n == 1:
return i
a *= base
if __name__ == "__main__":
max = (0, 0)
for i in range(2, 1000):
if tools.gcd(i, 10) == 1:
x = multiplicative_order(10, i)
if x > max[1]:
max = (x, i)
print max
def cont(self):
return gcd(*self.coef)
def depth_position_attribute(depth, position):
g = gcd(depth - position, position)
return (depth - position) / g, position / g
def depth_position_attribute(depth, position):
g = gcd(depth - position, position)
return (depth - position) / g, position / g
