Question: Given any integer $m\geq 1$, is it true that the interval $$\left[ \frac{-1+\sqrt{1+8m}}{2}, \frac{1+\sqrt{-7+8m}}{2} \right]$$ contains exactly one integer only?
From graphing, this seems to be true. However, I do not know how to prove it.
As the length of the interval is less than $1$, there can be at most one integer in the interval. We now show that there always exists one.
If there is no integer in the interval, then there is an integer $k$ such that $k< \frac{-1+\sqrt{1+8m}}{2}$ and $\frac{1+\sqrt{-7+8m}}{2} < k + 1$.
This is equivalent to $(2k + 1)^2 < 1 + 8m$ and $(2k + 1)^2 > -7 + 8m$.
However we know that $(2k + 1)^2$ is congruent to $1$ mod $8$, so it cannot lie between $1 + 8m$ and $-7 + 8m$.