A pentagonal square triangular number is a number that is simultaneously a pentagonal number P_l, a square number S_m, and a triangular number T_n. This requires a solution to the system of Diophantine equations 1/2 l(3l - 1) = m^2 = 1/2 n(n + 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 P_1 = S_1 = T_1 = 1, 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 10^22166 (E. W. Weisstein, Sept. 12, 2003).

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