Find all the prime numbers (such as 2, 3, 5, 7, 11, 13, 17, 19, …) between 1 and 4027. To print the primes, write a function which prints a list in a table which “reads down”, using as few rows as possible, and using as few sheets of paper as possible, given the number of columns for the table. Print this table to a text file. The parameters of the functions should be the list, the number of columns desired, and a file or file name, if used. (For the purposes of this project, open the file and print the table using an editor or word processor.) Each number should be right-justified in its column. The height of the columns should all be the same, except for perhaps the last column, which might have a few blank entries toward its bottom row. After printing the table of primes, print a second table (to a separate file) of the twin primes found among the original prime numbers, again reading down. Finally, print to the console the following statistics: number of primes found and the number of twin prime pairs found

Prime numbers are a critical concept in mathematics and have been a subject of great interest and study for centuries. In this assignment, we are tasked with finding all the prime numbers between 1 and 4027 and printing them in a table format with a specific number of columns. We are also required to create a separate table containing the twin prime numbers, and finally, provide statistics on the total number of primes found and the number of twin prime pairs.

To approach this problem, we need to devise an efficient algorithm for finding prime numbers within a given range. A common and effective method is the Sieve of Eratosthenes. This algorithm, named after the Greek mathematician Eratosthenes, allows us to systematically eliminate non-prime numbers from the list. The remaining numbers in the list after the elimination process are prime.

Let’s begin by implementing the Sieve of Eratosthenes algorithm to generate a list of prime numbers between 1 and 4027. We will use an array to represent the numbers within the given range, initializing all elements to true initially.

“`python
def find_primes(n):
primes = [True] * (n + 1)
primes[0] = primes[1] = False

p = 2
while p * p <= n: if primes[p]: for i in range(p * p, n + 1, p): primes[i] = False p += 1 prime_list = [] for num in range(2, n + 1): if primes[num]: prime_list.append(num) return prime_list primes = find_primes(4027) ``` Once we have obtained the list of prime numbers, we can proceed to print them in the table format specified in the assignment. We need to determine the number of rows required for the table, which can be calculated as the ceiling of the total number of primes divided by the number of columns desired. Then, we can iterate through the primes list and print them in the appropriate format. ```python def print_table(prime_list, columns, file_name): rows = (len(prime_list) + columns - 1) // columns with open(file_name, 'w') as file: for row in range(rows): for col in range(columns): index = row + col * rows if index < len(prime_list): file.write('{:>5}’.format(prime_list[index]))
file.write(‘n’)

print_table(primes, 10, ‘prime_table.txt’)
“`

In addition to the table of primes, we are required to create a separate table containing the twin prime numbers. Twin primes are pairs of prime numbers that differ by 2 (e.g., (3, 5) or (11, 13)). We can modify our previous function to identify and print the twin primes in a similar manner.

“`python
def find_twin_primes(prime_list):
twin_primes = []

for i in range(len(prime_list) – 1):
if prime_list[i + 1] – prime_list[i] == 2:
twin_primes.append((prime_list[i], prime_list[i + 1]))

return twin_primes

twin_primes = find_twin_primes(primes)
print_table(twin_primes, 10, ‘twin_prime_table.txt’)
“`

Finally, to provide the required statistics, we can simply print the length of the prime list and the number of twin primes found.

“`python
print(‘Number of primes found:’, len(primes))
print(‘Number of twin prime pairs found:’, len(twin_primes))
“`

Executing the above code will generate two separate text files containing the tables of primes and twin primes, respectively, as well as outputting the statistics to the console.