Unusual equation(s) obtained at an attempt to solve an open problem While trying to solve an open problem I arrived at the following equation:
$$56l^3+60l^2+10l+6=6\sum_{j=1}^{l+1}a_jb_j$$ , where $a_j,b_j$ are constrained by $1<a_j,b_j<4l$, for $j=1,...,l+1$
$a_j,b_j$ can be more precisely constrained but I will not go now into details because even this looks complicated enough.
Now, further constraints: $a_j$ and $b_j$ are also all odd and we have $a_j \neq b_j$ and we have that all $a_ib_i$ are different and we also must have $  4 | a_jb_j - a_ib_i$ and we must also have this extremely strong divisibility condition $ (i-j) | a_ib_i-a_jb_j$. 
Here we, in fact, have an infinite number of equations, because we get for every natural $l \in \mathbb N$ one non-linear Diophantine equation.
Honestly, I do not know who to consult when it comes to this problem.
What to do? 

Subtract the equation for $l-1$ from the equation for $l$ and divide by 6 to get $a_{l+1}b_{l+1}=28l^2-8l+1<(4l)^2=16l^2$, so $12l^2-8l+1=(2l-1)(6l-1)<0$, i.e. $1/6<l<1/2$. But since $l$ is an integer, the problem has no solutions.
Update: Apparently, now $a_i$’s and $b_i$’s may be different for different $l$, so this answer has been rendered obsolete.
