A hexagonal pentagonal number is a number which is simultaneously pentagonal and hexagonal. Let 
denote the 
th pentagonal number and
the 
th
hexagonal number for positive
integers 
and 
. A number which is both pentagonal and hexagonal satisfies
the equation 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Therefore, 
and 
satisfy the Pell-like equation
 |
(3)
|
The first few solutions with 
and 
are 
,
(5, 3), (19, 11), (71, 41), (265, 153), (989, 571), .... These give the solutions
, (1, 1), (10/3, 3), (12, 21/2), (133/3,
77/2), (165, 143), ..., of which the integer solutions
are (1, 1), (165, 143), (31977, 27693), (6203341, 5372251), ... (OEIS A046178
and A046179), corresponding to the pentagonal
hexagonal numbers 1, 40755, 1533776805, 57722156241751, ... (OEIS A046180).
