Can we define the Gaussian distribution by taking the power of 2 instead of the exponential function?

Question

The probability density function of the Gaussian distribution is $1 / {\sqrt{2 \pi \sigma^2} * e^{- {(x - \mu)^2}\over{2 \sigma^2}}}$. Can we instead define it as something like $1 / {\sqrt{2 \pi \sigma^2} * 2^{- {(x - \mu)^2}\over{2 \sigma^2}}}$ and adjust the normalization so that it integrates to 1?