A pentagonal square triangular number is a number that is simultaneously a pentagonal number 
, a square number 
, and a triangular
number 
. For positive integer
indices 
, 
,
and 
, this requires a solution to the system of Diophantine
equations
 |
(1)
|
Solutions of this system can be searched for by checking pentagonal triangular numbers (for which there is a closed-form
solution) up to some limit to see if any are also square.
Other than the trivial case 
,
using this approach shows that none of the first 9690 pentagonal
triangular numbers are square, thus showing that there is no other pentagonal
square triangular number less than 
(E. W. Weisstein, Sept. 12, 2003).
The only positive pentagonal square triangular number is 1 (Katayama et al. 2020, p. 4). J. Sillcox (pers. comm., Nov. 8, 2003 and Feb. 17, 2006)
noted the connection with the simultaneous Pell equations
considered by Anglin (1996). Setting 
, 
, and 
gives
 |  |  |
(2)
|
 |  |  |
(3)
|
Anglin (1996, pp. 355 and 359) determined the solutions for all 19900 pairs of integer coefficients 
in the simultaneous Pell
equations 
, 
.
For example, 
is a solution for 
and 
.
For 
and 
, the only positive solution is 
, yielding 
.
