Twin and Mersenne Primes

Sometimes I need to do some math that's above my students' levels, just for the sake of my own sanity. So sometimes I'll just ponder a puzzle and investigate it to see if I can find an answer. This is one of those times.

Twin primes are any two prime numbers that have a difference of two, such as 3 and 5 or 29 and 31.

Mersenne primes are prime numbers that are one less than a power of 2. In other words, they can be represented by the expression (2^p)-1, where p is a prime number. Examples include 3, 7, 31, etc.

You may have noticed that 3 is in both examples listed above. This brings the question:

Can you find another Mersenne prime that is
the smaller of two Twin Primes?

-- OR --

Can you explain why no other examples exist?