(Stanat and McAllister [1977]) In some computers, an integer (positive or negative) is represented by using bit strings of length \(p\) . The last bit in the string represents the sign, and the first \(p - 1\) bits are used to encode the integer. What is the largest number of distinct integers that can be represented in this way for a given \(p\)?

We can represent \(2^{p-1}\) integers and multiply that by 2 (sign: - or +) to get the total of \(2 \cdot 2^{p-1} = 2^p\) integers

What if 0 must be one of these integers? (The sign of 0 is + or -.)

We must exclude one of the two 0’s that we have included in the total above to get \(2^{p}-1\) integers