Pentagonal Triangular Number is a number which is simultaneously a pentagonal number $P_n$ and triangular number $T_m$ . Such numbers exist when
$$
\frac{1}{2}n(3n-1) = \frac{1}{2}m(m+1)
$$
This problem boils down to the pell-like quadratic Diophantine equation:
$$
x^2-3y^2 = -2
$$
Using Brahmagupta's Lemma:
$$
\text{Theorem 2.22 (Brahmagumpta)}\;\; \text{If }\; u^2 − Pv2 = k \;\; \text{ and } \;\; r^2 − Ps^2 = l \;\;\;\text{then} (ur ± Pvs)^2 − P(us ± vr)^2 = kl.
$$
One can prove that there are infinitely many $(n,m)$ such that $P_n = T_m$
(I did this around $3$ years back for extra credit in my number theory class)
I was just wondering if there were already papers or instances where somebody already showed this.
Thanks.
EDIT
I explicitly ask for papers here and not websites where they discuss the general topic. I'm looking for a formal proof of something I've already witnessed so that I may compare their methodology with mine.