【Abstract】In knot theory, the AJ conjecture states a close connection between the A-polynomial of a knot K and the q-recurrence equation that the colored Jones function of K satisfies. After introducing these concepts, we show that verifying the AJ conjecture for specific knots naturally leads to the problem of factoring q-shift operators. We discuss some improvements on existing factorization algorithms, their implementation and application to knots with few crossings and their connected sums.