Pentagonal Square Number
A number which is simultaneously a pentagonal number 
and a square number 
. Such numbers exist when
 |
(1)
|
Completing the square gives
 |
(2)
|
 |
(3)
|
 |
(4)
|
Substituting 
and 
gives the Pell equation
 |
(5)
|
which has solutions 
, (49, 20), (485, 198), .... In terms of 
, these give (1,1), (25/3, 10), (81, 99), (2401/3, 980), (7921, 9701), ..., of which the whole number solutions are 
, (81, 99), (7921, 9701), (776161, 950599), ... (OEIS A046172 and A046173), corresponding to the pentagonal square numbers 1, 9801, 94109401, 903638458801, 8676736387298001, ... (OEIS A036353).
REFERENCES:
Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall, 1996.
Sloane, N. J. A. Sequences A036353, A046172, and A046173 in "The On-Line Encyclopedia of Integer Sequences."
