Goldbach's Conjecture — Every Even Number Is the Sum of Two Primes

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> GOLDBACH'S CONJECTURE

goldbach difficulty: 1–4 also known as: Goldbach partition, strong Goldbach conjecture

The idea in plain English: Take any even number greater than 2. You can always find two prime numbers that add up to it. That's it. 4 = 2+2. 6 = 3+3. 8 = 3+5. 10 = 3+7 (or 5+5). No matter how large the even number, there's always at least one pair of primes that sum to it. Each such pair is called a Goldbach partition. The puzzle asks you to find one such partition for a given even number.

Why this really exists: The conjecture was first stated by Prussian mathematician Christian Goldbach in a 1742 letter to Leonhard Euler. Euler — one of history's greatest mathematicians — called it "completely certain" even though he couldn't prove it. Over 280 years later, it remains one of the most famous unsolved problems in number theory. It's been verified by computer for all even numbers up to 4 × 10¹⁸ (that's 4,000,000,000,000,000,000), but a general proof still eludes us. Understanding prime distribution is central to modern cryptography.

▸ Concrete Example: 10 and 28

Let's find Goldbach partitions for two numbers:

Even number = 10

Possible prime pairs (both must be prime):
2 + 8 → 8 is not prime ✗
3 + 7 → 3 is prime, 7 is prime ✓
5 + 5 → 5 is prime, 5 is prime ✓

Goldbach partitions of 10: (3, 7) and (5, 5)

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Even number = 28

Try candidate pairs:
3 + 25 → 25 not prime ✗
5 + 23 → both prime ✓
11 + 17 → both prime ✓

Goldbach partitions of 28: (5, 23) and (11, 17)

Notice that a number can have multiple valid partitions. The puzzle accepts any valid pair. The key insight: you only need to check primes up to n/2, because after that the pairs just reverse.

▸ How to Find a Goldbach Partition (Step by Step)

1. You are given an even integer n (n > 2)

2. Generate a list of all prime numbers up to n

3. For each prime p in the list (where p ≤ n/2):

4. Let q = n - p

5. Check if q is also prime

6. If yes, (p, q) is a valid Goldbach partition

7. Return the pair as your answer

# Python solver for finding a Goldbach partition:
def is_prime(x):
    if x < 2: return False
    for i in range(2, int(x**0.5) + 1):
        if x % i == 0: return False
    return True

def goldbach_partition(n):
    for p in range(2, n // 2 + 1):
        if is_prime(p) and is_prime(n - p):
            return (p, n - p)
    return None # never happens for n > 2

# Example:
# >>> goldbach_partition(100)
# (3, 97)

💡 For efficiency, pre-compute primes with the Sieve of Eratosthenes if your n is large. The puzzle numbers stay modest, but the pattern scales.

▸ Difficulty & Puzzle Patterns

Difficulty	What to Expect	Example n	Sample Partition
1	Small even numbers (4–30), partitions obvious	12	(5, 7)
2	Moderate numbers (up to 200), slightly larger primes	98	(19, 79)
3	Larger numbers (up to 2,000), primes less intuitive	1,000	(3, 997) or (59, 941)
4	Large range (up to 10,000+), requires prime detection	10,000	(3, 9997) or (59, 9941)

⚠️ Difficulty 5+ puzzles may use the weak Goldbach conjecture variant (odd numbers as sum of three primes), which was proven in 2013 by Helfgott.

▸ Real-World Connections

Prime distribution: Goldbach is deeply tied to the Riemann Hypothesis — understanding how primes are distributed is the key to both problems
Cryptography: RSA, ECC, and Diffie-Hellman all rely on the difficulty of prime-related problems. Research into Goldbach informs our understanding of prime density
Computational verification: Testing Goldbach up to 4×10¹⁸ drove innovations in distributed computing and primality testing algorithms
Additive number theory: Goldbach's conjecture is the flagship problem in additive number theory — the study of how numbers combine to form other numbers
Educational mathematics: An excellent introduction to primes, sieves, and the boundary between computational verification and mathematical proof

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