Pentagonal Square Number

Author

Eric W. Weisstein
March 25, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/PentagonalSquareNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PentagonalSquareNumber.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Computation

Simplified Pell

Reduce[(x^2)-6(y^2)==1&&0<x<1000&&y>0,{x,y},Integers]

Full

Reduce[PentagonalNumber[n]==SquareNumber[m]&&(m|n)∈Integers&&m>0&&n>0,{n,m},Integers]/.C[1]->k

soln=Table[
{-(((((5-2 (Sqrt[6])))^(1+2 k))-(((5+2 (Sqrt[6])))^(1+2 k)))/(4 (Sqrt[6]))),
(1/6) (1+(1/2) ((((5-2 (Sqrt[6])))^(1+2 k))+(((5+2 (Sqrt[6])))^(1+2 k))))},{k,0,10}]//Expand

SquareNumber/@First/@soln