2 added 379 characters in body

# 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.


# Jelly, 5 bytes

&‘<ÆP


Try it online!

# Jelly, 5 bytes

&‘<ÆP


Try it online!

### 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.

1

# Jelly, 5 bytes

&‘<ÆP


Try it online!