class TestSieve(unittest.TestCase):
def setUp(self):
self.primes = Primes(100)
def test_size(self):
self.assertEqual(self.primes.size, 100)
def test_isPrime_83(self):
self.assertEqual(self.primes.is_prime[83], True)
def test_first_5_primes(self):
self.assertEqual(self.primes.first(5), [2, 3, 5, 7, 11])
def test_primes_up_to_13(self):
self.assertEqual(self.primes.up_to(13), [2, 3, 5, 7, 11, 13])
def test_primes_up_to_100(self):
self.assertEqual(self.primes.up_to(100), [
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97
])
def test_primes_1000th_prime_is_7919(self):
s = Primes(10000)
self.assertEqual(s.first(1000)[-1], 7919)
def Euler_123(low = 10**10): ''' Returns the nth prime that has a remainder above low when using the given formula ''' p = Primes(10**6) primes = p.pList() for x in range(0,len(primes)): n = mpz(x+1) p = mpz(primes[x]-1)**n + mpz(primes[x]+1)**n if p % mpz(primes[x])**2 > low: return n
def pe41():
"""
>>> pe41()
7652413
"""
primes = Primes(1000000)
for perm in permutations(range(7, 0, -1)):
n = list_num(perm)
if primes.is_prime(n):
return n
return -1
def test_primes(self):
primes = Primes()
assert 0 not in primes
assert 1 not in primes
taketen = primes[1:11]
correct = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
taketen = list(taketen)
assert taketen == correct
for x in range(1, 11):
assert primes[x] == correct[x - 1]
assert 104744 not in primes
assert primes[10001] == 104743
itr = iter(primes)
for _ in range(10100):
next(itr)
assert list(primes.countdown(29))[::-1] == correct
def Euler_35(top=10**6): ''' Returns the number of circular primes below the top ''' def rotate(num): return int(str(num)[-1] + str(num)[0:-1]) count = 0 primes = Primes(top) primes.sieve() for val in primes.pList(): val2 = rotate(val) while val2 != val: if not primes.isPrime(val2): break val2 = rotate(val2) if val == val2: count += 1 return count
def pe27():
"""
>>> pe27()
(-59231, (-61, 971, 71))
"""
m = (0, 0, 0)
primes = Primes(15000)
for a in range(-999, 1000):
for b in range(max(2, 1 - a), 1000):
n, cnt = 0, 0
while 1:
p = n * (n + a) + b
if primes.is_prime(p):
cnt += 1
else:
break
n += 1
if cnt > m[2]:
m = (a, b, cnt)
return (m[0] * m[1], m)
def pe27():
"""
>>> pe27()
(-59231, (-61, 971, 71))
"""
m = (0, 0, 0)
primes = Primes(15000)
for a in range(-999, 1000):
for b in range(max(2, 1 - a), 1000):
n, cnt = 0, 0
while 1:
p = n * (n + a) + b
if primes.is_prime(p):
cnt += 1
else:
break
n += 1
if cnt > m[2]:
m = (a, b, cnt)
return (m[0] * m[1], m)
def generate_code(n, k):
primes = Primes(n + k)
primes_list = primes.primes_list
#primes_list=primes.primes_3mod4()
print(primes_list)
words = gen_words(primes_list[0:k])
primes = primes_list[k:n + k]
print(primes_list)
print(primes)
code = [[legendre_symbol(w, p) for p in primes] for w in words]
return code
def test_factors(self):
primes = Primes()
assert primes.factors(0) == []
assert primes.factors(1) == []
assert primes.factors(2) == [2]
assert primes.factors(6) == [2, 3]
assert primes.factors(8) == [2, 2, 2]
assert primes.factors(29) == [29]
def generate_jacobi_code(n, k):
nlog = int(math.log(n, 2))
primes = Primes(nlog + k + 1)
primes_list = primes.primes_list
#primes_list=primes.primes_3mod4()
print(primes_list)
words = gen_words(primes_list[0:k])
primes = primes_list[k:nlog + k]
encoders = gen_words(primes)
print(primes_list)
print(encoders)
code = [[jacoby_symbol(w, m) for m in encoders] for w in words]
return code
def Euler_131(top=6*10**3): ''' Returns the number of primes below the input where n**3+p*n**2 is a perfect cube ''' from primes import isPrime from primes import Primes foo = lambda p,n: n**3 + p * n**2 def bsearch(l, val, low, high): while low <= high: mid = (low + high) // 2 if val < l[mid]: high = mid - 1 continue elif val > l[mid]: low = mid + 1 continue elif val == l[mid]: return True return False cubelist = [x**3 for x in range(1,top)] cubetop = top a = Primes(top) count = 0 for p in a.pList(top): for n in range(1,top): tmp = foo(p,n) while tmp > cubelist[-1]: cubelist.append(cubetop**3) cubetop += 1 if bsearch(cubelist, tmp, 0, cubetop-1): print(p,n,tmp) count += 1 break return count '''
class TestSieve(unittest.TestCase):
def setUp(self):
self.primes = Primes(100)
def test_size(self):
self.assertEqual(self.primes.size, 100)
def test_isPrime_83(self):
self.assertEqual(self.primes.is_prime[83], True)
def test_first_5_primes(self):
self.assertEqual(self.primes.first(5), [2, 3, 5, 7, 11])
def test_primes_up_to_13(self):
self.assertEqual(self.primes.up_to(13), [2, 3, 5, 7, 11, 13])
def test_primes_up_to_100(self):
self.assertEqual(self.primes.up_to(100),
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89,
97])
def test_primes_1000th_prime_is_7919(self):
s = Primes(10000)
self.assertEqual(s.first(1000)[-1], 7919)
def pe49(limit=10000):
"""
>>> pe49()
(2969, 6299, 9629)
"""
pr = Primes()
for num in range(1111, limit):
sn = str(num)
if sn.find('0') >= 0: continue
if pr.is_prime(num):
pp = {num : 1}
for p in permutations(list(sn)):
n = int(''.join(p))
if pr.is_prime(n):
pp[n] = 1
primes = sorted(pp.keys())
pl = len(primes)
for a in range(pl):
if primes[a] == 1487: continue
for b in range(a+1, pl):
c = (primes[a] + primes[b]) >> 1
if c in pp:
return (primes[a], c, primes[b])
return None
def test_trunc_primes(self):
primes = Primes()
assert primes.truncatable(3797)
primes = Primes()
trunc = []
for prime in primes:
if primes.truncatable(prime):
trunc.append(prime)
if len(trunc) == 10:
assert sum(trunc) == 8920
break
def main():
a= Primes(2, 100000, 2000)
p = random.choice(a)
q = random.choice(a)
#print("随机生成两个素数p和q. p=",p," q=",q)
n = p * q
s = (p-1)*(q-1)
#print("The p is ", p)
#print("The q is ", q)
#print("The n(p*q) is ",n)
e = co_prime(s)
#e = 5
#print("根据e和(p-1)*(q-1))互质得到: e=", e)
d = find_d(e,s)
#print("根据(e*d) 模 ((p-1)*(q-1)) 等于 1 得到 d=", d)
#print("公钥: n=",n," e=",e)
#print("私钥: n=",n," d=",d)
pbvk = (n,e,d)
return pbvk
def test_is_circular(self):
primes = Primes()
assert primes.is_circular(2)
assert primes.is_circular(971)
assert not primes.is_circular(999953)
def primes():
return Primes()
import sys
sys.path.append('C:\\Users\\bob\\comp-sci\\project-euler\\toolkit')
from primes import Primes
p = Primes(1000000)
is_prime = p.is_prime
p_list = p.primes_list
def solve():
most_terms, prime = 0, 0
for terms in range(1, 547):
for i in range(len(p_list) - terms):
p_sum = sum(p_list[i:i+terms])
if p_sum > 1000000:
break
if is_prime(p_sum):
most_terms, prime = terms, p_sum
break
return prime
"""
project euler #77
matt donahoe
i enjoyed thinking of this one. did some back of the envelope (literally)
discovered a dynamic programming solution
"""
from primes import Primes
p=Primes(5000)
ways = {} #ways.get((n,p),0) = number of ways to sum to n using only primes >= p
primes = []
n=1
while s<=50000000:
n+=1
s = 0
if p.is_prime(n):
s = 1
primes.append(n)
for x in primes[::-1]:
s+=ways.get((n-x,x),0)
ways[(n,x)] = s
print n,s
def empty_Primes():
from primes import Primes
new_primes = Primes()
return new_primes
def test_prime_100th_element():
"""Test method first returns correct element."""
from primes import Primes
Primes = Primes()
assert Primes.first(100)[99] == 541
def test_primes_slice():
"""Test method first returns correct slice."""
from primes import Primes
Primes = Primes()
assert Primes.first(20)[-5:] == [53, 59, 61, 67, 71]
def test_primes_1000th_prime_is_7919(self):
s = Primes(10000)
self.assertEqual(s.first(1000)[-1], 7919)
def button_handle_random(self, event): # 按钮(随机素数)事件
a = Primes(2, 5000, 1000)
p = random.choice(a)
q = random.choice(a)
self.content_P.set("%s" % p)
self.content_Q.set("%s" % q)
#!/usr/bin/python3
'''
Primes generator using Sieve of Eratosthenes
Use -l to limit output or use Ctrl+C to interrupt.
'''
import argparse
import sys
from primes import Primes
argument_parser = argparse.ArgumentParser('Primes generator')
argument_parser.add_argument('-l', dest='limit', type=int, default=None)
limit = argument_parser.parse_args().limit
prime_numbers = Primes()
for prime in prime_numbers:
if limit and prime > limit:
sys.exit()
print(prime)
def ListOfPrimes(num):
listofPrimenumbers = []
for i in range(2, num):
if Primes(i):
listofPrimenumbers.append(i)
return listofPrimenumbers
def test_primes_1000th_prime_is_7919(self):
s = Primes(10000)
self.assertEqual(s.first(1000)[-1], 7919)
#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
What is the smallest odd composite
that cannot be written as the sum of a prime and twice a square?
"""
from primes import Primes
_SQ = [i * i << 1 for i in range(100)]
primes = Primes()
def is_goldbach(n):
res = False
for i in range(100):
t = n - _SQ[i]
if t < 2:
break
if primes.is_prime(t):
res = True
break
return res
def pe46():
"""
>>> pe46()
5777
"""
limit = 10000
def setUp(self):
self.primes = Primes(100)
def test_prime_80thth_element():
"""Test method first returns correct element."""
from primes import Primes
Primes = Primes()
assert Primes.first(80)[79] == 409
def setUp(self):
self.primes = Primes(100)
def test_totient(self):
correct = [(2, 1), (3, 2), (4, 2), (5, 4), (6, 2), (7, 6), (8, 4),
(9, 6), (10, 4), (4079147, 4074720)]
primes = Primes()
for (k, v) in correct:
assert primes.totient(k) == v
from primes import Primes testP = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997] print Primes.first(168) == testP print Primes.first(1000)[-1] == 7919
#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
Solve project euler 58
Investigate the number of primes
that lie on the diagonals of the spiral grid.
"""
from primes import Primes
primes = Primes(30000)
def pe58(p=0.1):
"""
>>> pe58()
26241
"""
l = 7
ps = 8
d = 49
ds = 13
while ps > p * ds:
l += 2
i = l - 1
ps += primes.is_prime(d + i) + primes.is_prime(
d + 2 * i) + primes.is_prime(d + 3 * i)
d += i << 2
ds += 4
return l
import timeit
from triangular import count_factors
from triangular import gen_triangular_number
from primes import Primes
i = 1
while True:
i +=1
pobj = Primes(i)
t = gen_triangular_number(i)
factors = count_factors(pobj, t)
if factors >= 500:
print ("factors", factors, "t", t)
break
""" The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ? """ from primes import Primes primes = Primes(10000) print(list(primes.prime_factors((600851475143))))
def test_consecutive_sum_max(self):
primes = Primes()
assert (2 == primes.consecutive_sum_max(5))
assert (0 == primes.consecutive_sum_max(11))
assert (6 == primes.consecutive_sum_max(41))
from primes import Primes
n = i = 1
p = Primes()
percent = 1
diagonals = 1 # this takes into account the center (1)
prime_diagonals = 0
while percent > 0.1:
for x in range(4):
n+=2*i
prime_diagonals += p.is_prime(n)
diagonals += 4
i+=1
percent = float(prime_diagonals) / diagonals
print prime_diagonals, '/', diagonals, ',', percent
print 'side =', int(n**(0.5)), 'primes =', prime_diagonals, '/', diagonals, percent
def test_consecutive_sum_max_length(self):
primes = Primes()
assert primes.consecutive_sum_max_length(50) == (41, 6)
'''
import maths
primesd = maths.primesd(10**6)
primes = [x for x in primesd.keys() if primesd.get(x) is 1]
'''
from primes import Primes
primes = Primes(10**6)
ps = primes.pList()
def Euler_50():
high = 0
for i in range(10):
sums = 0
x = 0
if i % 10000 == 0:
print('new', i)
for j in ps[i:]:
x += 1
sums += j
if sums > 10**6: break
if primes.isPrime(sums):
if x > high:
high = x
maxs = sums
#;print('high:',high,'sum:',sums,'i:',primes.pList(i))
return maxs
if __name__ == '__main__':
print(Euler_50())
'''
import maths
primesd = maths.primesd(10**6)
primes = [x for x in primesd.keys() if primesd.get(x) is 1]
'''
from primes import Primes
primes = Primes(10**6)
ps = primes.pList()
def Euler_50():
high = 0
for i in range(10):
sums = 0
x = 0
if i%10000==0:
print('new',i)
for j in ps[i:]:
x+=1
sums +=j
if sums>10**6:break
if primes.isPrime(sums):
if x>high:high = x;maxs=sums
#;print('high:',high,'sum:',sums,'i:',primes.pList(i))
return maxs
if __name__=='__main__':
print(Euler_50())
# Prints the time needed to covers the onion shell
ts2 = time.time()
print (a, ts2-ts)
ts = ts2
N = 1000 # N is half the side
L = 2*N-1 # L is the side of the square
# Creates a matrix, initialized to 0
# Each elements will represent the number of primes generated
matrix = [[0]*L for i in range(L)]
# Primes generator
primes = Primes()
# For the moment the maximum sequence has 0 length
# Each time we find a longer sequence we will record it here
# together with its parameters (point in the square, ...)
maxi = {'n': 0}
for a, b in get_coeffs(N):
i = a + N - 1 # indices in the matrix
j = b + N - 1
f = make_fun(a, b)
# how many primes does f produces?
n = 0
while True:
# Prints the time needed to covers the onion shell
ts2 = time.time()
print(a, ts2 - ts)
ts = ts2
N = 1000 # N is half the side
L = 2 * N - 1 # L is the side of the square
# Creates a matrix, initialized to 0
# Each elements will represent the number of primes generated
matrix = [[0] * L for i in range(L)]
# Primes generator
primes = Primes()
# For the moment the maximum sequence has 0 length
# Each time we find a longer sequence we will record it here
# together with its parameters (point in the square, ...)
maxi = {'n': 0}
for a, b in get_coeffs(N):
i = a + N - 1 # indices in the matrix
j = b + N - 1
f = make_fun(a, b)
# how many primes does f produces?
n = 0
while True:
