Is it a Mersenne Prime?

05AB1E, 5 bytes

A positive number in the form 2ⁿ - 1 in binary only consists of 1's.

Code:

b`¹pP

Explanation:

b`      # Push each digit of the binary representation of the number onto the stack
  ¹p    # Check if the input is prime
    P   # Take the product of all these digits

Uses the CP-1252 encoding. Try it online! or Verify all test cases.

Jelly, 5 bytes

&‘<ÆP

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How it works

&‘<ÆP  Main link. Argument: x

 ‘     Yield x+1.
&      Take the bitwise AND of x and x+1.
       This yields 0 iff x is a Mersenne number, i.e., iff x+1 is a power of 2.
   ÆP  Yield 1 if x is a prime, 0 if not.
  <    Compare the results to both sides,
       This yields 1 iff x is both a Mersenne number and a prime.

Python, 45 bytes

lambda n:-~n&n<all(n%i for i in range(2,n))<n

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How it works

The three terms of the chained comparison

-~n&n<all(n%i for i in range(2,n))<n

do the following:

• -~n&n computes the bitwise AND of n + 1 and n. Since n consists solely of 1 bits if it is a Mersenne number, the bitwise AND will return 0 if (and only if) this is the case.

• all(n%i for i in range(2,n)) returns True if and only if n mod i is non-zero for all values of i in [2, …, n - 1], i.e., if and only if n has no positive divisors apart from 1 and n.

  In other words, all returns True if and only if n is a composite number, i.e., n is either 1 or a prime.

• n is self-explanatory.

The chained comparison returns True if and only if the individual comparisons do the same.

• Since all returns either True/1 or False/0, -~n&n<all(n%i for i in range(2,n)) can only return True if -~n&n yields 0 (i.e., if n is a Mersenne number) and all returns True (i.e., if n either 1 or a prime).

• The comparison all(n%i for i in range(2,n))<n holds whenever n > 1, but since all returns True if n = 1, it does not hold in this case.