Find all possible pairs (x, y) of prime numbers, with x less than or equal to y, and each of x and y being less than 2023, such that:

• y divides x2 + 8 and,
• x divides y2 + 8

Provide valid reasoning as to why there are no further solutions.

*** Extra credit for an analytical solution.

^Note: This puzzle is adapted from a problem appearing in the Zhautykov Olympiad.