Python: Accept an even number from the user and create a combinations that express the given number as a sum of two prime numbers

Last update on December 21 2024 07:19:32 (UTC/GMT +8 hours)

Goldbach Partition Counter

Write a Python program which accepts an even number (>=4, Goldbach number) from the user and creates combinations which express the given number as a sum of two prime numbers. Print the number of combinations.

Goldbach number: A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes.[4] Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.
The expression of a given even number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some even numbers:
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7 = 5 + 5
12 = 7 + 5
...
100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53

Visual Presentation:

Sample Solution:

Python Code:

# Import necessary modules
import sys
from bisect import bisect_right
from itertools import chain

# Print statement to prompt the user to input an even number (0 to exit)
print("Input an even number (0 to exit):") 

# Set the upper bound for prime number computation
ub = 50000

# Initialize a boolean list to track prime numbers using the Sieve of Eratosthenes algorithm
is_prime = [0, 0, 1, 1] + [0]*(ub-3)
is_prime[5::6] = is_prime[7::6] = [1]*int(ub/6)

# Initialize a list to store prime numbers
primes = [2, 3]
append = primes.append

# Sieve of Eratosthenes algorithm to find prime numbers
for n in chain(range(5, ub, 6), range(7, ub, 6)):
    if is_prime[n]:
        append(n)
        is_prime[n*3::n*2] = [0]*((ub-n)//(n*2))
primes.sort()

# Infinite loop to continuously accept user input until 0 is entered
for n in map(int, sys.stdin):
    # Check if the entered number is 0, and break the loop if true
    if not n:
        break
    
    # Check if the number is odd
    if n % 2:
        # Print statement to display the number of combinations
        print("Number of combinations:")
        # Print the number of combinations using the count of is_prime[n-2]
        print(is_prime[n-2])
    else:
        # Print statement to display the number of combinations
        print("Number of combinations:")
        # Print the number of combinations using the count of prime pairs that sum to n
        print(len([1 for p in primes[:bisect_right(primes, n/2)] if is_prime[n-p]]))

Sample Output:

Input an even number (0 to exit):
 100
Number of combinations:
6

Explanation:

Here is a breakdown of the above Python code:

• First the code initializes the upper bound for prime number computation (ub).
• It initializes a boolean list (is_prime) using the Sieve of Eratosthenes algorithm.
• The code finds prime numbers and stores them in the list (primes).
• It enters an infinite loop to continuously accept user input until 0 is entered.
• Inside the loop, it checks if the entered number is odd and prints the number of combinations using is_prime[n-2].
• If the number is even, it prints the number of combinations using the count of prime pairs that sum to 'n'.

Flowchart:

Python Code Editor:

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