A Diophantine Problem Concerning Polygonal numbers

Abstract

Abstract. Motivated by some earlier Diophantine works on triangular numbers by Ljunggren and Cassels, we consider similar problems for general polygonal numbers.

On power values of pyramidal numbers, I

“…For the remaining newforms in Table 2, we find that for any prime q ∤ 2D that we test, p | Norm(q + 1 − c q (f )). (11) This suggests that the representation ρ f,p is reducible, which would be a contradiction. We proceed by applying [6, Proposition 2.2] to the newform f .…”

“…Pyramidal numbers are a special type of figurate number with many interesting properties and a rich history. The properties of figurate numbers, and in particular their relationship with perfect powers, have received much attention in the literature, see [8,11,15] for example, and the references therein. This paper is the second in a series of two papers.…”

Special formulas involving polygonal numbers and Horadam numbers

“…In [3] some summation formulas for polygonal numbers are derived. For some other interesting properties of P (r) n we refer the reader to [2,9,11,15,18]. Recall that the general Horadam sequence {w n } = {w n (a, b; p, q)} is a second order linear recurrence w n = pw n−1 − qw n−2 , n ≥ 2, with nonzero constant p, q and initial values w 0 = a, w 1 = b.…”

A Unique Perfect Power Decagonal Number

“…Polygonal numbers have been studied since antiquity [6, pages 1-39] and relations between different polygonal numbers and perfect powers have received much attention (see, for example, [7] and the references cited therein). Kim et al [7,Theorem 1.2] found all solutions to the equation P s (n) = y m when m > 2 and s ∈ {3, 5, 6, 8, 20} for integers n and y. We extend this result (for m > 1) to the case s = 10, that of decagonal numbers.…”

“…The n th s -gonal number, with , which we denote by , is given by the formula Polygonal numbers have been studied since antiquity [6, pages 1–39] and relations between different polygonal numbers and perfect powers have received much attention (see, for example, [7] and the references cited therein). Kim et al [7, Theorem 1.2] found all solutions to the equation when and for integers n and y . We extend this result (for ) to the case , that of decagonal numbers.…”