The New Year Coincidence?

Since we are given that $2021^{2020}\equiv 25 \pmod{31}$ and $2021\equiv 6 \pmod{31}$ , $2021^{2022}\equiv 2021^{2020}\cdot 2021^{2}\equiv 25\cdot 6^2\equiv 900\equiv 1 \pmod{31}$ Using the fact that $2022\equiv 7\pmod{31}$ , $2022^{2021}\equiv 7^{2021}\pmod{31}$ Note that $31$ is prime. By Fermat's Little Theorem , we have, $7^{30}\equiv 1\pmod{31}$ , $7^{2021}\equiv(7^{30})^{67}\cdot 7^{11}\equiv 7^{11}\equiv (7^3)^3\cdot 7^2\equiv 2^3\cdot 7^2\equiv 20 \pmod{31}$

Therefore, $2021^{2022}\equiv 1\pmod{31}$ and $2022^{2021}\equiv 20\pmod{31}$ which implies that they are not equivalent.