Determine all possible pairs (A,B) of positive integers that satisfy this equation:

• B^A2 = A^B+2

Provide valid reasoning for your answer.

Notes:
(i) x^x^x is equal to x^(x^x) and NOT (x^x)^x.

(ii) Computer program/excel solver assisted solutions are welcome, but a semi-analytic (hand calculator and p&p) methodology is preferred.

(iii) Adapted from a problem appearing in Taiwan M.O: 2000.