REPEATED BINOMIAL COEFFICIENTS AND HIGH-DEGREE CURVES

摘要

We consider the problem of characterizing solutions in (x, y) to the equation x y = xa y+b in terms of a and b. We obtain one simple result which allows the determination of a ratio in terms of a and b which the ratio x y must approximate. We then use a version of Siegel’s theorem on integral points to prove that in the case a 6= b, solutions to x y = xa y+b are finite. Finally, we make some observations about the potential utility of equations of the form x y = xa y+b in proving Singmaster’s conjecture, which is the main unsolved problem in the area of repeated binomial coecient study. We remark that this approach to the conjecture is markedly di?erent from previous approaches, which have only established logarithmic bounds on a function which counts the number of representations of t as a binomial coecient.