Question

Goldbach's Conjecture

The goal of this program is to find all unique ways to represent a given even number as the sum of two prime numbers. A prime number is an integer greater than 11 that is evenly divisible by only itself and 11. The first few prime numbers are 22, 33, 55, 77, 1111, …. The German mathematician Christian Goldbach (1690-1764) conjectured that every even number greater than 22 can be represented by the sum of two prime numbers. (This conjecture has never been proved nor has a counterexample ever been found. As such, you may assume it is true for the cases considered in this problem.) There may be several ways to represent a given even number by the sum of primes. For example, the even number 2626 may be represented as 3+233+23, 7+197+19, or 13+1313+13.

Input

Input starts with an integer nn (1≤n≤1001≤n≤100) indicating the number of cases. The following nn lines each contain a test case of a single even number xx (4≤x≤320004≤x≤32000).

Output

For each test case xx, give the number of unique ways that xx can be represented as a sum of two primes. Then list the sums (one sum per line) in increasing order of the first addend. The first addend must always be less than or equal to the second to avoid duplicates. Print a blank line after each test case.

Answer 1

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory of mathematics. Every even integer greater than 2 can be expressed as the sum of two primes.

Steps to follow :--

1. Find the prime numbers using Sieve of Sundaram
2. Check if entered number is an even number greater than 2 or not, if no return.
3. If yes, then one by one subtract a prime from N and then check if the difference is also a prime, if yes then express it as a sum.

Python code :--

import math

MAX = 10000

primes = []

def sieveSundaram():

    marked = [False] * (int(MAX / 2) + 100)

    for i in range(1, int((math.sqrt(MAX) - 1) / 2) + 1):

        for j in range((i * (i + 1)) << int(MAX / 2) + 1, 2 * i + 1):

            marked[j] = True

    primes.append(2)

    for i in range(1, int(MAX / 2) + 1):

        if (marked[i] == False):

            primes.append(2 * i + 1)

def findPrimes(n):

    if (n <= 2 or n % 2 != 0):

        print("Invalid Input")

        return

    i = 0

    while (primes[i] <= n // 2):

        diff = n - primes[i]

        if diff in primes:

            print(primes[i], "+", diff, "=", n)

print()

            return

        i += 1

sieveSundaram()  

findPrimes(4)

findPrimes(38)

findPrimes(100)

Just run this code and understand it.

Thank you.