Inspiration

• If n<0 and remains as an integer, the result of 2^n + 2 (let's called the sum: x) will be 2<x<3; therefore, n cannot be less than 0. If n>0 and remains as an integer, the result of 2^n +2 will always be an even number because you'll be multiplying an even number (2) by itself which will always have an even outcome. With all considered, 0 is the only possibility that will satisfy the conditions; and if one plug 0 in for n, the sum of the expression will be equal to 3, which is an odd number.

Relevant wiki: Parity of Integers

Since adding 2 to any number doesn't change it's parity, we have to find the smallest integer for $2^{x}$ is odd. Since fractions can't be even or odd, that means we have to find a positive integer. Since 2x, where x is an positive integer, is always even, we can rule out all integers. However, there is one exception: 0. Since $x^{0}$ is always equal to 1, that means that 0 is the only possible solution.

Does 0 count as an integer...?