Find odd numbers $(o_1,o_2,o_3,o_4)$ such that $o_1^2-o_2^2=2(o_3^2-o_4^2)$ such that $o_1>o_2$ and $o_3>o_4$ I am working on a graph labeling problem and am stuck at the following problem on odd numbers.
Find (all) odd numbers $(o_1,o_2,o_3,o_4)$ such that $o_1^2-o_2^2=2(o_3^2-o_4^2)$ such that $o_1>o_2$ and $o_3>o_4$?
Ideally I would like to prove that no such 4-tupule $(o_1,o_2,o_3,o_4)$ exists. However if they exist, I want to count, for given $n$, how many such 4-tupules exists such that the largest odd number i.e. $o_1 \leq n$.
My approach: Say the 4-tupule is $(2p+1,2q+1,2r+1,2s+1)$. The above equation simplifies to $(p-q)(p+q+1) = 2(r-s)(r+s+1)$. I am stuck here..
Thanks for going through this.

Given the condition, let use write
\begin{eqnarray}
o_1 &=& o,\\
o_2 &=& o - 4p,\\
o_3 &=& o - 2q,\\
o_4 &=& o - 2 r.
\end{eqnarray}
Then we get

$$
o\Big(p+q-r\Big)=2p^2+q^2-r^2.
$$
Case 1
When
$$
p+q-r = 0,
$$
we obtain
$$
p \Big( p -2q \Big) = 0,
$$
so we obtain
\begin{eqnarray}
o_1 &=& 1 + 2 v + 8 w,\\
o_2 &=& 1 + 2 v,\\
o_3 &=& 1 + 2 v + 6 w,\\
o_4 &=& 1 + 2 v + 2 w.
\end{eqnarray}
This gives for example
$$
\begin{array}{cc|cccc}
v & w & o_1 & o_2 & o_3 & o_4\\
\hline
0 & 1 & 9 & 1 & 7 & 3\\
0 & 2 & 17 & 1 & 13 & 5\\
0 & 3 & 25 & 1 & 19 & 7\\
1 & 1 & 11 & 3 & 9 & 5\\
1 & 2 & 19 & 3 & 15 & 7\\
1 & 3 & 27 & 3 & 21 & 9\\
2 & 1 & 13 & 5 & 11 & 7\\
2 & 2 & 21 & 5 & 17 & 9\\
2 & 2 & 29 & 5 & 23 & 11\\
\end{array}
$$
Case 2
When
$$
p+q-r \ne 0,
$$
we can write
$$
r = p + q - a,
$$
so we obtain
$$
a o = 2 p^2 + q^2 - \Big( p + q - a \Big)^2
= p \Big( p - 2q \Big) - a^2  + 2ap + 2aq,
$$
whence
$$
p = ak,
$$
thus
$$
o = a \Big( k^2 + 2k - 1 \Big) - 2 \Big( k - 1 \Big) q.
$$
However, $o$ is odd and therefore $a$ is odd and $k$ is even, whence
$$
o = \Big(2 u + 1 \Big) \Big( 4 v^2 + 4 v - 1 \Big) - 2 \Big( 2 v - 1 \Big) w.
$$
So we obtain
\begin{eqnarray}
o_1 &=& \Big(2 u + 1 \Big) \Big( 4 v^2 + 4 v - 1 \Big) - 2 \Big( 2 v - 1 \Big) w,\\
o_2 &=& \Big(2 u + 1 \Big) \Big( 4 v^2 - 4 v - 1 \Big) - 2 \Big( 2 v - 1 \Big) w,\\
o_3 &=& \Big(2 u + 1 \Big) \Big( 4 v^2 + 4 v - 1 \Big) - 2 \Big( 2 v + 1 \Big) w,\\
o_4 &=& \Big(2 u + 1 \Big) \Big( 4 v^2 - 4 v - 1 \Big) - 2 \Big( 2 v + 1 \Big) w.
\end{eqnarray}
