A generalization of Gosper’s algorithm to bibasic hypergeometric summation. An algebraically motivated generalization of Gosper’s algorithm to indefinite bibasic hypergeometric summation is presented. In particular, it is shown how Paule’s concept of greatest factorial factorization of polynomials can be extended to the bibasic case. It turns out that most of the bibasic hypergeometric summation identities from the literature can be proved and even found in this way. A Mathematica implementation of the algorithm is available from the author.