Possibility of the result of length of two n-bits length integer I have known from my calculation and Maximum bits of two n-bits ingeter that the maximum bits of two n-bits integer is 2n bits. I need to handcraft a RSA and I know that the modulus which is the key length needs to be 1024 bits to 2048 bits. From the result above, I know that the maximum bits of two n-bits integer is 2n bits, **but there is possibility that the length of the result is only 2n-1 bits.** So I would like to know the possibility of the result of length of two n-bits length integer when 1. both two integer is greater than $2^{n-2}$ 2. both two integer is in $[1,2^{n-1}-1]$ A proof will be appreciated.

I know this is not what you asked, but here is an easy way to achieve what you want:

First, generate random pairs of 16-bit integers $p_0$ and $q_0$ (each $\ge 2^{15}$), until you get a pair that satisfies

$$p_0\cdot q_0\ge 2^{31}$$

Then generate two $n$-bit primes $p$ and $q$ with most significant 16 bits $p_0$ and $q_0$, respectively. These will satisfy $pq>2^{n-1}$, which is what you want.

The probability that such a randomly generated pair satisfies $p_0\cdot q_0\ge 2^{31}$ is approximately $2(1-\ln 2)\approx 0.614$, so the first stage shouldn't take long.