There is an ancient problem, I remember reading once in a book while I was a kid, that says: there was a father who had $N$ sons and $T$ cows. He divided the cows among the sons in the following order: for the first son, $1 + 1/7$ of the remaining; for the second son $2+ 1/7$ of the remaining; ..., for the $i$'th son $i+1/7$ of the remaining... the question asked for the numbers T and N. In fact $N=6$ and $T=36$ is a solution. I found that it can be generalized to any given ratio $1/M$ and $N=M-1$ and $T=N^2$ is a solution. I was wondering if there is any other solution to the general problem in $\mathbb{Z}$. I have formalized the problem as following:
We have a system of equations described by the following $N+1$ equations defined in $\mathbb{Z}$: \begin{align*} n_1 &= 1+ \frac{T-1}{M}\\ n_2 &= 2+ \frac{T-2-n_1}{M}\\ &\vdots\\ n_i &= i+ \frac{T-i-\sum\limits_{j=1}^{i-1} n_j}{M}\\ T&=\sum\limits_{i=1}^{N} n_i. \end{align*}
One set of solutions is given by: \begin{align*} T &= N^2\\ n_i&=N\\ M&=N+1 \end{align*}
For example, when $N=6$, $T=36$, $n_1 = n_2 = \cdots = n_6 = 6$ is a solution.
I'm wondering if there is any other non-trivial integer solutions to the above system of equations.
Thanks,
MG