TITLE: A conjecture about triangular numbers
QUESTION [7 upvotes]: There is one and only one pair of natural numbers $m$ and $n$ ($15$ and $21$) such that: $m$ and $n$ are triangular numbers; the sum of their squares is a triangular number ($666$); and the sum of the triangular numbers $m(m+1)/2$ and $n(n+1)/2$ is a triangular number $k(k+1)/2$ ($k=26$). I have tested pairs of numbers from $m+n=2$ to $m+n=100 000 000$ using a computer program. (Notice that $15, 20, 25$ is a Pythagorean triple, giving the sides of a triangle that's similar to the sacred Egyptian triangle $3, 4, 5$.) I know, $666$ is the number of the beast and all that stuff, but this is a serious mathematical problem.

REPLY [2 votes]: You know what else is crazy about this?
We see that even $(15 + 21 = 36)$ is also a triangular number so for the $n$th triangular number formula $\frac{n(n+1)}{2}$, we see that :-
$1$. $15$ is triangular with $n=5$, $21$ is triangular with $n=6$. So, as a previous comment noted, $15$ and $21$ are consecutive triangular numbers.
$2.$ $15^2 + 21^2 = 666$ is triangular with $n=36$ .
$3.$ $\frac{15(15+1)}{2}+\frac{21(21+1)}{2} = 351$ is triangular with $n=26$.
$4.$ $15+21=36$ is triangular with $n=8$ (actually, the sum of any two consecutive triangular numbers is always a perfect square. what’s up with a triangular square number showing up here?)
What are the connections between these values of $n$? What makes $15$ and $21$ ($n=5$ and $n=6$) so special? And of course $666$ is thrown into the mix here too.
Does it mean something deeper in regards to our universe? Probably not. But this is really interesting.