A188896 Numbers n such that there is no square n-gonal number greater than 1.

10, 20, 52, 164, 340, 580, 884, 1252, 1684, 2180, 2740, 4052, 4804, 5620, 6500, 7444, 8452, 9524, 10660, 11860, 13124, 14452, 15844, 17300, 18820, 20404, 22052, 25540, 27380, 29284, 31252, 33284, 35380, 37540, 39764, 42052, 44404, 46820, 49300, 51844

Offset: 1

Author

T. D. Noe, Apr 13 2011

Keywords

• nonn

Comments

It is easy to find squares that are triangular, pentagonal, hexagonal, etc. So it is somewhat surprising that there are no square 10-gonal numbers other than 0 and 1. For these n, the equation 2*x^2 = (n-2)*y^2 - (n-4)*y has no integer solutions x>1 and y>1.

Chu shows how to transform the equation into a generalized Pell equation. When n has the form 2k^2+2 (A005893), then the Pell equation has only a finite number of solutions and it is simple to select the n that produce no integer solutions greater than 1.

The general case is in A188950.

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..244
• Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781-802.

Crossrefs

Cf. A001107 (10-gonal numbers), A051872 (20-gonal numbers), A188892, A100252, A188950, A005893.

Subsequence of A271624. - Muniru A Asiru, Oct 16 2016

Programs

Mathematica

P[n_,k_]:=1/2n(n(k-2)+4-k); data1=2#^2+2&/@Range[2,161]; data2=Head[Reduce[m^2==P[n,#] && 1Ant King, Mar 01 2012 *)