Floor function 2

Now consider putting two consecutive positive integers as a difference of two squares so that the difference is 104, if not around 104. This is also represented as ${ (n+1) }^{ 2 }-{ n }^{ 2 }=104$ .

and we get $2n+1=104\\ n=51.5$

But $n$ is not an integer, so let $n=52$ .

Now subbing in $n=52$ gives us ${ (n+1) }^{ 2 }-{ n }^{ 2 }=105>104$

And $\frac{105}{104}>1$ and this means that the floor function of any term from $n=52$ will have a difference of at least one and thus the terms after $n=52$ are distinct.

And thus, every term before $n=52$ may or may not be distinct. The common difference of each term, $2n+1 <104$ . But $\frac{52^2}{104}=26$ . So this means there are $26-0+1=27$ distinct terms in the range $1\le n\le 52$ .

Now for the range $53\le n\le 103$ , since the common difference of $2n+1>104$ , each of the term in the range $53\le n\le 103$ is distinct, so there are $103-53+1=51$ distinct terms in the range $53\le n\le 103$ .

Thus, there are $27+51=78$ distinct terms in all.