Step by Step solution about How to resolve the algorithm Twin primes step by step in the Python programming language

This Python code defines a function count_twin_primes(limit) that calculates the number of twin prime pairs (pairs of prime numbers that differ by 2) less than a given limit. The code also includes a test function to demonstrate the usage of count_twin_primes for various limits.

A step-by-step explanation of the code:

1. Global Variable primes:
   • A global variable primes is defined as a list containing the initial prime numbers up to 19.
2. count_twin_primes Function:
   • This function takes a positive integer limit as its parameter.
   • It calculates the number of twin prime pairs less than the given limit.
   • The function first checks if the given limit exceeds the largest prime number in the primes list. If so, it extends the list of primes up to the given limit or a reasonable limit based on memory constraints.
   • It then iterates through the extended primes list and uses a list comprehension to filter out numbers that are not prime by marking them in a sieve list.
   • The function appends the non-marked numbers (i.e., prime candidates) to the primes list.
   • Finally, it counts the number of twin prime pairs by checking if the difference between two consecutive prime numbers in the primes list is 2.
3. test Function:
   • This function tests the count_twin_primes function for different limits.
   • It prints the number of twin prime pairs less than each specified limit.
4. Usage of test Function:
   • The test function is called with various limits, such as 10, 100, 1000, 10000, 100000, 1000000, 10000000, and 100000000.
   • It demonstrates the calculation of twin prime pairs for each limit.

In summary, this code calculates the number of twin prime pairs up to a given limit using the Sieve of Eratosthenes algorithm to find prime numbers. It extends the list of prime numbers on demand as needed to handle larger limits.

primes = [2, 3, 5, 7, 11, 13, 17, 19]


def count_twin_primes(limit: int) -> int:
    global primes
    if limit > primes[-1]:
        ram_limit = primes[-1] + 90000000 - len(primes)
        reasonable_limit = min(limit, primes[-1] ** 2, ram_limit) - 1

        while reasonable_limit < limit:
            ram_limit = primes[-1] + 90000000 - len(primes)
            if ram_limit > primes[-1]:
                reasonable_limit = min(limit, primes[-1] ** 2, ram_limit)
            else:
                reasonable_limit = min(limit, primes[-1] ** 2)

            sieve = list({x for prime in primes for x in
                          range(primes[-1] + prime - (primes[-1] % prime), reasonable_limit, prime)})
            primes += [x - 1 for i, x in enumerate(sieve) if i and x - 1 != sieve[i - 1] and x - 1 < limit]

    count = len([(x, y) for (x, y) in zip(primes, primes[1:]) if x + 2 == y])

    return count


def test(limit: int):
    count = count_twin_primes(limit)
    print(f"Number of twin prime pairs less than {limit} is {count}\n")


test(10)
test(100)
test(1000)
test(10000)
test(100000)
test(1000000)
test(10000000)
test(100000000)