This is my first posting here, so I am not sure this is in the correct place.
I have this math problem: Find the number of distinct integers in this set:
$$\left\{\left[\frac{1^2}{5012}\right], \left[\frac{2^2}{5012}\right],\ldots, \left[ \frac{5012^2}{5012}\right]\right\},$$
where $[x]$ denotes the greatest integer less than or equal to $x$.
I am not very familiar with the mathematical notation involved with this problem, so I wrote a computer program that solves it (in Java)
I get the correct answer: 3760, and I get the entire set of distinct integers. But I do not know how to represent this mathematically. Any help is appreciated! Thank you.
There are two regimes here. We know the gap between squares grows as the square root grows larger. As long as the gap between $k^2$ and $(k+1)^2$ is less than $5012$ we will hit every integer because the gaps between the fractions is less than $1$. When the gap is larger than $5012$, the integers will all be distinct. As the gap between $k^2$ and $(k+1)^2$ is $2k+1$, it becomes larger than $5012$ at $k=2506$. So in the set we have every integer from $0$ through $\frac {2506^2}{5012}=1253$, or $1254$ of them, and then all the ones from $\lfloor \frac {2507^2}{5012}\rfloor$ through $\lfloor \frac {5012^2}{5012}\rfloor$, or $2506$ of them, a total of $3760$.