I was reading a paper about solutions to $f(y) = P(m)$, where $f(y) \in \mathbb Z[y]$ and $P(m) = n(n + 1) \ldots (n + m - 1)$ is a product of $m$ consecutive integers.
On the first page, it is mentioned that it is not difficult to get the solutions in the case that $f(y)$ is irreducible. However, I was confused about two things.
- First, the author mentions a condition on $f(y)$ which implies that the equation $f(y) = P(m)$ has no solution, but doesn't mention when a solution does exist.
- Secondly, I'm not sure how to generalize the method used in the example in the paper. In the example given, the number of cases where it is possible to have a solution is quite small, so it isn't hard to check what the solutions are (if any).
I think this might be a problem when $f(y)$ has large degree. Also, I tried to solve $f(y) = P(m)$ when $f(y) = py + 1$, where $p$ is an odd prime. Using the condition given, I found that the smallest prime $q$ which does not divide $f(y)$ for any $y$ is $p$. So, there are no solutions when $m > p$. But I'm not sure how to start finding general solutions (or finding conditions for $m$ when solutions do exist).
Is there something I'm misinterpreting in the paper? Any suggestions on how to start or other references to look at?
Note: This is cross-posted from math.SE: https://math.stackexchange.com/questions/419918/solutions-to-fy-nn-1-ldots-n-m-1