Can we find an integer $m$ such that: $2^{2p-2}-2^{p}+3=m²$

Question

Let $p$ a prime number. Can we find an integer $m$ such that: $$2^{2p-2}-2^{p}+3=m²$$

Answer 1

Hint: $2^{2p-2}-2^{p}+3=m²=(2^{p-1}-1)^2+2$ see this mod 4 we have a contraddition.

Answer 2

Hints:

$$2^{2p-2}-2^{p}+3=(2^{p-1})^2-2\cdot 2^{p-1}+3=(2^{p-1}-1)^2+2=m²$$

Does there exist two elements $s$ and $t$ from $\Bbb Z$ such that $s^2-t^2=2$?