This question is a formulation of Goldbach's Conjecture, one of the most famous unsolved problems in number theory. The conjecture suggests that:

Every even integer greater than 2 can be expressed as the sum of two prime numbers.

For example:

• $4 = 2 + 2$
• $6 = 3 + 3$
• $8 = 3 + 5$
• $10 = 3 + 7$
• $12 = 5 + 7$

This has been checked for very large numbers, and no counterexamples have been found. However, as of now, it remains unproven for all integers, despite extensive computational verification and partial results in number theory.

Current Status:

• Verified: It has been verified for even numbers up to $4 \times 10^{18}$ using computer algorithms.
• Unproven: A formal proof that Goldbach's Conjecture holds for all even integers greater than 2 is still missing.

Conclusion:

Though no counterexample has been found, the question of whether every even integer greater than 2 can always be written as the sum of two primes remains an open problem in mathematics.

Would you like more information on this conjecture or number theory in general? Here are a few related questions:

1. What methods are used to verify Goldbach’s conjecture for large numbers?
2. What are some famous unsolved problems in mathematics besides Goldbach’s conjecture?
3. How does the distribution of prime numbers affect conjectures like Goldbach’s?
4. How does the computational verification of conjectures work?
5. What’s the relationship between Goldbach's conjecture and the twin prime conjecture?

Tip: If you're diving into number theory, try looking into the Prime Number Theorem, which gives an approximation for the number of primes less than a given number. It’s a great stepping stone for deeper exploration into prime number distribution.