A number which is simultaneously a pentagonal number P_n and a square number S_m. Such numbers exist when 1/2 n(3n - 1) = m^2. Completing the square gives 1/2 n(3n - 1) = 3/2(n^2 - 1/3 n) = 3/2 (n - 1/6)^2 - 3/72 = m^2 3/2 (6n - 1)^2 - 3/2 = 36m^2 (6n - 1)^2 - 24m^2 = 1.

pentagonal number | pentagonal square triangular number | square number