  We know that Khayyam wrote a treatise, now lost, called Problems of Arithmetic involving the determination of n-th roots (Youschkevitch and Rosenfeld 1973 ). In his Algebra Khayyam writes that methods for calculating square and cube roots come from India, and that he has extended them to the determination of roots of any order. Even more interestingly, he says that he has demonstrated the validity of his methods using proofs that “are purely arithmetic, founded on the arithmetic of the Elements.” If both of these statements are true, then it is hard to avoid the conclusion that Khayyam had within his power the binomial theorem  (a + b)n = an + nan−1b + … + bn, which would be the earliest appearance of this important result in medieval Islam.