A number which is simultaneously pentagonal and hexagonal. Let 
denote the 
th pentagonal number and
the 
th
hexagonal number, then a number which is both
pentagonal and hexagonal satisfies the equation 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Therefore, defining
 |  |  |
(3)
|
 |  |  |
(4)
|
gives the Pell-like equation
 |
(5)
|
The first few solutions 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).
