Regular Polygonal Numbers and Generalized Pell Equations

Abstract. In the eighteenth century, both square numbers and triangular num-bers were investigated by Euler and Goldbach (1742), who determined the recurrence relations satisfied by the sequence and established the general formulae explicitly. It seems to the author that the topics around this subject have not been touched in mathematical literature. As the first attempt to explore it, this work will present a systematic procedure to deal with the problem. For the regular (λ, μ)-polygonal numbers, the corresponding Diophantine equations will be reduced to the general-ized Pell equations. Then solutions of the associated Pell equations will essentially enable us to resolve the problem. By means of Computer Algebra, the recurrence relations and generating functions satisfied by (λ, μ)-polygonal numbers can be re-covered systematically. As exemplification, the results on the first twenty regular (λ, μ)-polygonal sequences will be presented in details