Is the sum of two prime numbers always a prime number?

Concept: A universal claim of the form 'X is always true' can be disproved by producing just one counterexample — you don't need to check every case to show it fails somewhere. Separately, recall a parity fact about primes: 2 is the only even prime, so every other prime is odd. Adding two odd numbers always gives an even result, and if both primes are greater than 2, that even sum exceeds 2 — so it must be composite, not prime.

Application: Take the primes 3 and 7 (both odd, since neither is 2). Their sum is 3 + 7 = 10. Since 10 is even and greater than 2, it fits the parity rule above and is indeed composite (10 = 2 × 5). This one case already breaks the claim that the sum of two primes is always prime, so the answer must be 'No'.

Cross-check: Try the pair 2 and 3 instead — their sum is 2 + 3 = 5, which IS prime. This does not contradict the reasoning above: 2 is the exceptional even prime, so the parity argument (odd + odd = even greater than 2) only applies when both chosen primes are greater than 2. When 2 is one of the two primes, the sum can occasionally be prime. So the sum is sometimes prime (e.g., 2 + 3 = 5) and sometimes not (e.g., 3 + 7 = 10) — confirming it is not ALWAYS prime.

Conclusion: Because a counterexample exists (3 + 7 = 10), the statement 'the sum of two primes is always prime' is false. The correct answer is No.