No, the number 5 is prime, yet in binary it is 101. The new prime number could have many zeroes interspersed with ones, so you don't know how many 1 digits it has.

EDIT: I have realized my mistake in the representation of 2's complement. However, I will leave this here for posterity.

It's a Mersenne Prime, so it is 1 less than some power of 2. This results in all of its non-leading digits in binary being 1s.

But 5 is not 2^n-1

5 is a prime number true, but not a Mersenne prime number. The Mersenne prime number 31 has 5 as its exponent n [1].

[1] http://mathworld.wolfram.com/MersennePrime.html