Find a formula for all pentagonal numbers which are also square numbers.

Question

I can get the formula for n-th pentagonal number is $P_n=\frac{3n^2-n}{2}$, but I do not know how to get the formula which is also square number. enter image description here

Answer 1

Hint: $3n^2 - n = 2 m^2$ can be written as $(6n-1)^2 - 1 = 24 m^2$. The Diophantine equation $x^2 - 1 = 24 m^2$ is a Pell equation.