What properties does the ratio of two triangular numbers $M$ and $N$ have, assuming $N \mid M$?

Question

This is a follow-up question to this MSE post.

Call a number $X$ triangular if it could be written in the form $x(x+1)/2$ where $x$ is a positive integer.

Here is my question in this post:

What properties does the ratio of two triangular numbers $M$ and $N$ have, assuming $N \mid M$?

That is, if $$M = \dfrac{m(m+1)}{2}$$ and $$N = \dfrac{n(n+1)}{2}$$ where $m$ and $n$ are positive integers, then what can be said about $$\dfrac{M}{N} = \dfrac{\dfrac{m(m+1)}{2}}{\dfrac{n(n+1)}{2}} = q,$$ where $q$ is a positive integer?

Please note that this is not a homework question.

MY ATTEMPT

The equation can be rewritten (without fractions) as $$m(m+1)=qn(n+1),$$ which can then be treated as a quadratic in $m$ $$m^2 + m - qn(n+1) = 0$$ and as a quadratic in $n$ $$qn^2 + qn - m(m+1) = 0.$$

The first quadratic (in $m$) has a positive integer solution if $$1 + 4qn(n+1)$$ is a perfect square. The second quadratic (in $n$) has a positive integer solution if $$q^2 + 4qm(m+1)$$ is a perfect square.

This gives rise to the Diophantine system of equations $$\begin{cases} 1 + 4qn(n+1) = r^2, r \in \mathbb{Z} \\ q^2 + 4qm(m+1) = s^2, s \in \mathbb{Z} \end{cases}.$$

Subtracting the first equation from the second, we get $$(q^2 - 1) + \bigg(4qm(m+1) - 4qn(n+1)\bigg) = s^2 - r^2,$$ which gives $$(q-1)(q+1) + 4q(m-n)(m+n+1) = (s-r)(s+r).$$

From the first equation, I have $$r^2 \equiv 1 \pmod 4$$ so that $r$ is odd.

From the second equation, I have $$s^2 \equiv 0 \pmod q$$ and $$s^2 \equiv q^2 \pmod 4,$$ so that $s$ and $q$ have the same parity.

Alas, there is where I get stuck.

Can any more additional information be obtained about $m$, $n$, and $q$?

Answer 1

$m(m+1)=qn(n+1)\implies (2 m + 1)^2 - q (2 n + 1)^2 = 1 - q$.

Smallest solutions for primes $q<256$:

q = 2    (m,n) = (3, 2)
q = 3    (m,n) = (2, 1)
q = 5    (m,n) = (5, 2)
q = 7    (m,n) = (6, 2)
q = 11    (m,n) = (21, 6)
q = 13    (m,n) = (77, 21)
q = 17    (m,n) = (84, 20)
q = 19    (m,n) = (455, 104)
q = 23    (m,n) = (45, 9)
q = 29    (m,n) = (29, 5)
q = 31    (m,n) = (3471, 623)
q = 37    (m,n) = (185, 30)
q = 41    (m,n) = (5535, 864)
q = 43    (m,n) = (9675, 1475)
q = 47    (m,n) = (140, 20)
q = 53    (m,n) = (4080, 560)
q = 59    (m,n) = (1770, 230)
q = 61    (m,n) = (3949139, 505635)
q = 67    (m,n) = (175473, 21437)
q = 71    (m,n) = (12921, 1533)
q = 73    (m,n) = (8604875, 1007124)
q = 79    (m,n) = (395, 44)
q = 83    (m,n) = (332, 36)
q = 89    (m,n) = (2108499, 223500)
q = 97    (m,n) = (290, 29)
q = 101    (m,n) = (909, 90)
q = 103    (m,n) = (1268342, 124973)
q = 107    (m,n) = (5456, 527)
q = 109    (m,n) = (160775, 15399)
q = 113    (m,n) = (1695, 159)
q = 127    (m,n) = (29021024, 2575199)
q = 131    (m,n) = (55413, 4841)
q = 137    (m,n) = (32558735, 2781680)
q = 139    (m,n) = (418446990, 35492210)
q = 149    (m,n) = (10430, 854)
q = 151    (m,n) = (11481993341, 934391366)
q = 157    (m,n) = (130781, 10437)
q = 163    (m,n) = (377019489, 29530445)
q = 167    (m,n) = (1001, 77)
q = 173    (m,n) = (519, 39)
q = 179    (m,n) = (30125699, 2251700)
q = 181    (m,n) = (5302214, 394110)
q = 191    (m,n) = (57652776, 4171608)
q = 193    (m,n) = (965, 69)
q = 197    (m,n) = (2561, 182)
q = 199    (m,n) = (122864568110, 8709638309)
q = 211    (m,n) = (1882488227916, 129595834148)
q = 223    (m,n) = (1560, 104)
q = 227    (m,n) = (1589, 105)
q = 229    (m,n) = (182054, 12030)
q = 233    (m,n) = (7648542579, 501072684)
q = 239    (m,n) = (44789555, 2897195)
q = 241    (m,n) = (240, 15)
q = 251    (m,n) = (30948048, 1953423)