Question

How to prove that the mersenne number is not psuedoprime to the base 3 ??

Answer 1

Start by eliminating composite exponents p of 2p−1, so then only prime exponents are left. If 2p−1 is composite with p prime, then every prime divisor q of 2p−1 has the form 2kp+1. If the divisor q of 2p−1 does not divide 3p−1, then 2p−1 is not a strong pseudoprime base 3. Do this for all divisors q of 2p−1.