On power values of pyramidal numbers, I

Abstract

For m ≥ 3, the mth order pyramidal number is defined by x(x + 1)((m−2)x+5−m)/6. These combinatorial numbers play an important role in number theory and discrete mathematics. For m = 3 and 4, all the power values of these polynomials have been already found. In the present paper the authors deal with square values for larger m. All integer solutions of the corresponding elliptic curves are given for 3 ≤ m ≤ 100, m = 5 (Proposition 1). The main result is the resolution, for a conjecturally infinite sequence of integers m, of the corresponding elliptic diophantine equation (Theorem 2). 2010

Equal values of pyramidal numbers

“…where Poly m (x) denotes the sequence of polygonal numbers (for details please refer to [BPT98]) and proved that for all but a finite, computable set of pairs (m, n), max(x, y) is effectively bounded. In 2012, Dujella, Győry and Pintér [DGP12] studied the power values of pyramidal numbers. Recently, in two papers Pintér and Varga [PV11] and Hajdu, Tengely, Pintér and Varga [HPTV14] used various effective methods to investigate the equal values of general figurate numbers.…”