# Weakly NP-complete problems and strongly NP-complete problems

I'm still confused with the terms Weakly NP-complete and strongly NP-complete. Many people states that weakly NP-complete problems are the problems with time complexity growing as a polynomial in the magnitude of the input, while strongly NP-complete problems are the problems with time complexity growing as a polynomial in the size of the input.
Does this make checking whether a number is prime or not (which is always considered as polynomial-time problem) a weakly NP-complete problem since its time complexity is growing as a polynomial in the magnitude of the input but as an exponential in the size of the input?

2. Unary encoding, in which (ignoring signs) $n \geq 0$ is encoded as $0^n1$.