Hướng dẫn find large prime numbers in python - tìm số nguyên tố lớn trong python

Question

Để xác định xem một số là một số nguyên tố, có một sàng và các bài kiểm tra nguyên thủy.

Nội dung chính Show

Làm thế nào để bạn tìm thấy số nguyên tố lớn nhất trong Python?
Làm thế nào để bạn tìm thấy các số nguyên tố lớn?
Làm thế nào để bạn kiểm tra một số lượng lớn là số nguyên tố hay không trong Python?
Làm thế nào để bạn tạo ra các số nguyên tố lớn trong Python?

# for large numbers, xrange will throw an error.
# OverflowError: Python int too large to convert to C long
# to get over this:

def mrange(start, stop, step):
    while start < stop:
        yield start
        start += step

# benchmarked on an old single-core system with 2GB RAM.

from math import sqrt

def is_prime(num):
    if num == 2:
        return True
    if (num < 2) or (num % 2 == 0):
        return False
    return all(num % i for i in mrange(3, int(sqrt(num)) + 1, 2))

# benchmark is_prime(100**10-1) using mrange
# 10000 calls, 53191 per second.
# 60006 function calls in 0.190 seconds.

Đây dường như là nhanh nhất. Có một phiên bản khác sử dụng not any mà bạn thấy,

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

Tuy nhiên, trong các điểm chuẩn, tôi đã nhận được 70006 function calls in 0.272 seconds. về việc sử dụng all 60006 function calls in 0.190 seconds. trong khi kiểm tra xem

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

0 có phải là chính không.

Nếu bạn cần tìm Prime cao nhất tiếp theo, phương pháp này sẽ không phù hợp với bạn. Bạn cần phải đi với một bài kiểm tra nguyên thủy, tôi đã tìm thấy thuật toán Miller-Rabin là một lựa chọn tốt. Nó chậm hơn một chút phương pháp Fermat, nhưng chính xác hơn so với giả hành. Sử dụng phương pháp đã đề cập ở trên mất +5 phút trên hệ thống này.

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

1 Thuật toán:

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

2 Algoithm:

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

Để có được số nguyên tố cao nhất tiếp theo:

def next_prime(num):
    if (not num & 1) and (num != 2):
        num += 1
    if is_prime(num):
        num += 2
    while True:
        if is_prime(num):
            break
        num += 2
    return num

print next_prime(100**10-1) # returns `100000000000000000039`

# benchmark next_prime(100**10-1) using Miller-Rabin algorithm.
1000 calls, 337 per second.
258669 function calls in 2.971 seconds

Sử dụng thử nghiệm

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

2, chúng tôi đã đạt điểm chuẩn là

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

4, nhưng bạn có cơ hội giả giả cao hơn.

Vì vậy, nếu chỉ cần kiểm tra xem một số là số nguyên tố hay không, phương pháp đầu tiên cho

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

5 hoạt động tốt. Đó là nhanh nhất, nếu bạn sử dụng phương thức

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

6 với nó.

Lý tưởng nhất, bạn sẽ muốn lưu trữ các số nguyên tố được tạo ra bởi

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

7 và chỉ cần đọc từ đó.

Ví dụ: sử dụng

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

7 với thuật toán

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

1:

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

Bạn sẽ không thể làm điều này với

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

0 một cách kịp thời. Tôi thậm chí không thể làm điều đó trên hệ thống cũ này.

Và để chắc chắn rằng

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

1 và

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

1 mang lại một giá trị chính xác, điều này cũng đã được kiểm tra bằng cách sử dụng các thuật toán

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

2 và ____24.

Xem: Fermat.py, Miller_Rabin.py và Solovay_strassen.py trên gist.github.com

Chỉnh sửa: Đã sửa lỗi trong

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

7

Làm thế nào để bạn kiểm tra một số lượng lớn là số nguyên tố hay không trong Python?

Để tìm một số nguyên tố trong Python, bạn phải lặp lại giá trị từ đầu đến cuối bằng cách sử dụng vòng lặp và cho mỗi số, nếu nó lớn hơn 1, hãy kiểm tra xem nó có chia n không. Nếu chúng ta tìm thấy bất kỳ số nào khác phân chia, in giá trị đó.

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

1

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

2

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

4

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

6

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

8

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2not any0

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2not any2

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2not any4

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2not any6

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2not any8

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

0

Làm thế nào để bạn tạo ra các số nguyên tố lớn trong Python?

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

1all8

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

260006 function calls in 0.190 seconds.0

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

260006 function calls in 0.190 seconds.2

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

260006 function calls in 0.190 seconds.4

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

260006 function calls in 0.190 seconds.6

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

260006 function calls in 0.190 seconds.8

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

00

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

0

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

1

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

03

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

05

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

07

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

09

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

11

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

13

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

15

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

0

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

1

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

18

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

20

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

22

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

24

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

26

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

28

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

30

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

0

Để tạo một nguyên tố, trước tiên chúng ta tạo một số nguyên ngẫu nhiên trong phạm vi (2K-1,2K), thì các quy tắc sau được áp dụng:.

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

1

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

48

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

50

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

52

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

54

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

56

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

58

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

60

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

0

print next_prime(10^301)

# prints in 2.9s on the old single-core system, opposed to fermat's 2.8s
1000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000531

1

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

63

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

65

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

67

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

69

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

71

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

73

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

75

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

2

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

77

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

78

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

79

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

80

Các

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

79

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

02

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

81

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

04

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

05

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

06

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

08

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

10

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

12

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

13

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

14

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

15

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

17

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

08

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

19

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

13__

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

32

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

33

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

35

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

36

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

82

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

38

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

79

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

40

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

81

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

42

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

23

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

81

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

46

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

48

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

88

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

89

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

81

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

04

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

46

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

19

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

1

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

23

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

06

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

61

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

89

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

42

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

91

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

89

Các

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

81

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

79

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

86

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

17

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

89

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

90

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9____29

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

89

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

06

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

82

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

97

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

12

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

00

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

14

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

02

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

03

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

17

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

89

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

07

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

1__

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

32

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

82

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

97

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

82

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

05

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

81

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

27

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

29

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

81

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

12

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

00

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

14

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

02

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

35

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

37

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

83

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

1

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

41

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

17

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

86

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

82

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

97

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

12

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

00

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

14

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

02

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

03

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

17

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

89

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

07

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

1__

def is_prime(num)
    # ...
    return not any(num % i == 0 for i in mrange(3, int(sqrt(num)) + 1, 2))

81

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

04

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

05

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

06

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

08

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

9

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

10

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

12

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

13

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

14

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

15

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

17

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

08

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

19

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

13__

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

74

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

07

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

35

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

06

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

79

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

80

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

81

def is_prime(num):
    if num == 2:
        return True
    if not num & 1:
        return False
    return pow(2, num-1, num) == 1

82

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

16

from random import randrange
def is_prime(n, k=10):
    if n == 2:
        return True
    if not n & 1:
        return False

    def check(a, s, d, n):
        x = pow(a, d, n)
        if x == 1:
            return True
        for i in xrange(s - 1):
            if x == n - 1:
                return True
            x = pow(x, 2, n)
        return x == n - 1

    s = 0
    d = n - 1

    while d % 2 == 0:
        d >>= 1
        s += 1

    for i in xrange(k):
        a = randrange(2, n - 1)
        if not check(a, s, d, n):
            return False
    return True

33

Làm thế nào để bạn tìm thấy số nguyên tố lớn nhất trong Python?

Báo cáo vấn đề. Cho một số nguyên dương n. ....

Approach..

Thí dụ. Nhập bản demo trực tiếp Math def MaxPrimeFactor (n): # Số phải thậm chí trong khi N % 2 == 0: MAX_PRIME = 2 N /= 1 # Số phải là lẻ cho I trong phạm vi (3, int (Math. ...

Đầu ra. ....

Conclusion..

Làm thế nào để bạn tìm thấy các số nguyên tố lớn?

Vì vậy, làm thế nào để tạo ra số nguyên tố lớn?..

Tạo một ứng cử viên chính. Giả sử chúng tôi muốn một số nguyên tố 1024 bit. Bắt đầu bằng cách tạo 1024 bit ngẫu nhiên. ....

Kiểm tra nếu số được tạo là số nguyên tố với Miller-Rabin. Chạy bài kiểm tra nhiều thời gian để làm cho nó hiệu quả hơn ..

Nếu số không phải là số nguyên tố, hãy khởi động lại từ đầu ..

Làm thế nào để bạn kiểm tra một số lượng lớn là số nguyên tố hay không trong Python?

Để tìm một số nguyên tố trong Python, bạn phải lặp lại giá trị từ đầu đến cuối bằng cách sử dụng vòng lặp và cho mỗi số, nếu nó lớn hơn 1, hãy kiểm tra xem nó có chia n không.Nếu chúng ta tìm thấy bất kỳ số nào khác phân chia, in giá trị đó.iterate the value from start to end using a for loop and for every number, if it is greater than 1, check if it divides n. If we find any other number which divides, print that value.

Làm thế nào để bạn tạo ra các số nguyên tố lớn trong Python?

Để tạo một nguyên tố, trước tiên chúng ta tạo một số nguyên ngẫu nhiên trong phạm vi (2K-1,2K), thì các quy tắc sau được áp dụng:..

Số (n) phải là> = 3.....

Làm một chút và (N & 1).....

Kiểm tra xem n%p là 0 (nói cách khác, n không chia hết bởi p) cho tất cả các số nguyên tố

Cuối cùng chúng tôi đạt được bài kiểm tra cốt lõi: Rabin-Miller ..

programming python RSA prime numbers

Hướng dẫn find large prime numbers in python - tìm số nguyên tố lớn trong python

Làm thế nào để bạn tìm thấy số nguyên tố lớn nhất trong Python?

Làm thế nào để bạn tìm thấy các số nguyên tố lớn?

Làm thế nào để bạn kiểm tra một số lượng lớn là số nguyên tố hay không trong Python?

Làm thế nào để bạn tạo ra các số nguyên tố lớn trong Python?

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