[nanis]

1 is 2 to the power 0 ... 0b0001

shifted left once, it becomes 2 to the power 1 ... 0b0010

shifted left twice, it becomes 2 to the power 2 ... 0b0100

shifted left three times, it becomes 2 to the power 3 ... 0b1000

etc until

shifted left 136_279_841 times, it becomes 2 to the power 136_279_84 ... 0b1000...many zeros...0000

subtract 1, it becomes

0b0111...many ones...1111

[schoen]

One funny thing about Mersenne primes is that, as a result of what you describe, they are exactly those primes whose binary representation consists of a prime number of ones!

The smallest Mersenne prime, three, is binary 11, while the next largest is seven (111), then 31 (11111), then 127 (1111111). The next candidate, 2047 (11111111111), is not prime.