Is there a mathematical formula for the nearest-square function?

Question

Let $x$ be a positive integer.

Is there a mathematical formula for $$f(x)=\text{nearest square to } x \text{ }(\text{in terms of } x)?$$

I tried searching for related questions in MSE and found this one, but my inquiry is not covered there.

I also tried searching via Google and found this closely related question in StackOverflow.

I used the search string "nearest-square function".

player3236 · Accepted Answer · 2020-11-11T07:25:17.623

4

By virtue of the accepted answer in your linked question, you can take the nearest integer function of $\sqrt x$ and square it. However it may not be obvious (or seemingly too obvious to be true) so here are more details:

The nearest integer function (for $x \notin \{k+\frac12| k \in \mathbb Z\}$ to avoid ambiguities) is equal to $\left\lfloor x + \frac12\right\rfloor$. Fortunately for the square root, it is never an integer plus half. Hence we can also write

$$f(x) = \left\lfloor \sqrt x + \frac12\right\rfloor^2$$

To prove that this formula works, we consider the cases (where $n \in \mathbb N \cup \{0\}$):

Case 1: $n^2 \le x \le n^2+n < (n+\frac12)^2$

Case 2: $(n+\frac12)^2< n^2+n+1 \le x < (n+1)^2$

and we see that both cases work perfectly fine, given that $x$ is a positive integer.

edited Nov 11 '20 at 07:25

answered Nov 11 '20 at 07:05

player3236

16,600

@nicomezi I think I may have made a mistake, however. $x=12.3$ is a counterexample. I need to rethink this – player3236 Nov 11 '20 at 07:08
1

@nicomezi Found the "fix" (there is no fix, since $x$ is an integer) – player3236 Nov 11 '20 at 07:16
Ho my gosh, I am terrible at math and reading now. Tired of myself, like really. Nice work btw – nicomezi Nov 11 '20 at 07:18
@nicomezi It was I who suggested $12.3$ in the first place, I would say the fault/oversight is mine. – player3236 Nov 11 '20 at 07:19
This is fantastic, @player3236. Just what I needed! Thanks. =) – Jose Arnaldo Bebita Dris Nov 11 '20 at 07:27
It turns out that what I actually need is the nearest square to, but larger than $x$. Does your formula remain valid, @player3236? Or will there be some adjustments? Please advise. – Jose Arnaldo Bebita Dris Nov 11 '20 at 14:33
1

Nearest square to, but larger... wouldn't that be rounding up, just like your linked question? – player3236 Nov 11 '20 at 14:41
Okay got it! Thanks again. =) – Jose Arnaldo Bebita Dris Nov 12 '20 at 08:25

Is there a mathematical formula for the nearest-square function?

1 Answers1

Linked