Mersenne check — Find the smallest positive prime p for which 2^p - 1 is not prime (i.e., not a Mersenne prime).

Introduction / Context:
Numbers of the form 2^p - 1 with p prime are called Mersenne numbers. Some are prime (Mersenne primes), but not all. The question asks for the smallest prime exponent p where 2^p - 1 fails to be prime. This evaluates your familiarity with early Mersenne values and quick compositeness checks.

Given Data / Assumptions:

• We consider prime exponents p in increasing order.
• We check whether 2^p - 1 is prime.
• We stop at the first counterexample.

Concept / Approach:
List early cases: p = 2 → 3 (prime), p = 3 → 7 (prime), p = 5 → 31 (prime), p = 7 → 127 (prime). The next prime exponent is p = 11, giving 2^11 - 1 = 2047. Factor 2047 to test primality.

Step-by-Step Solution:
Compute 2^11 - 1 = 2047.Check small prime divisors: 2047 ÷ 23 = 89 → exact factorization 2047 = 23 * 89.Therefore 2^11 - 1 is composite. No smaller prime exponent p gave a composite earlier.

Verification / Alternative check:
Prior exponents: 2 → 3, 3 → 7, 5 → 31, 7 → 127, each prime. Hence p = 11 is indeed the smallest prime producing a composite 2^p - 1.

Why Other Options Are Wrong:

• 5 / 17 / 29 / 7: For 5 and 7, 2^p - 1 is prime; 17 and 29 are larger than the smallest counterexample.

Common Pitfalls:
Assuming every 2^p - 1 with p prime is prime; not checking 2047; overlooking small factors such as 23 and 89.

Final Answer:
11