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Numbers n such that there is no square n-gonal number greater than 1.
12

%I #45 Nov 28 2017 20:12:12

%S 10,20,52,164,340,580,884,1252,1684,2180,2740,4052,4804,5620,6500,

%T 7444,8452,9524,10660,11860,13124,14452,15844,17300,18820,20404,22052,

%U 25540,27380,29284,31252,33284,35380,37540,39764,42052,44404,46820,49300,51844

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

%C 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.

%C 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.

%C The general case is in A188950.

%H Muniru A Asiru, <a href="/A188896/b188896.txt">Table of n, a(n) for n = 1..244</a>

%H Wenchang Chu, <a href="http://www.m-hikari.com/imf-password2007/13-16-2007/chuIMF13-16-2007.pdf">Regular polygonal numbers and generalized Pell equations</a>, Int. Math. Forum 2 (2007), 781-802.

%t P[n_,k_]:=1/2n(n(k-2)+4-k); data1=2#^2+2&/@Range[2,161]; data2=Head[Reduce[m^2==P[n,#] && 1<m && 1<n && !m==n,{m,n},Integers]]&/@data1; data3=Flatten[Position[data2,Symbol]]; data1[[#]]&/@data3 (* _Ant King_, Mar 01 2012 *)

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

%Y Subsequence of A271624. - _Muniru A Asiru_, Oct 16 2016

%K nonn

%O 1,1

%A _T. D. Noe_, Apr 13 2011