# Closed-form for Floor Sum 3 - With knowledge of inner expression

Consider the following sum: $$\sum_{k=0}^{n}\left\lfloor\sqrt{k^2+N}\right\rfloor$$

Assume we know the factorization of $N$, in other words, we know for which $k$, $k^2+N$ will be square, according to Fermat's Factorization Method.

$N$ in this case is a positive odd integer.

Can we use this knowledge to get a closed-form for the sum?

Thanks a lot

How large is $n$ compared to $N$? Is $n\approx\sqrt{N}$? You are counting the lattice points in a region bounded by a hyperbola, and $\sqrt{x^2+N}$ behaves like $x$ for large values of $x$. Additionally, it is not likely that your expression has a nice closed form, but it is for sure pretty close to $$\int_{0}^{n}\sqrt{x^2+N}\,dx = \frac{1}{4} \left(2 n \sqrt{n^2+N}-N \log N+2 N\log\left(n+\sqrt{n^2+N}\right)\right)$$ which if $n=\sqrt{N}$ behaves like $\left(\sqrt{2}+\log(1+\sqrt{2})\right)\frac{N}{2}\approx\frac{7}{6}N$ for large values of $N$.
Also notice that the problem of finding the first two terms of the asymptotic expansion of $$\sum_{k=0}^{\sqrt{N}}\left\lfloor\sqrt{N\color{red}{-}k^2}\right\rfloor$$ as $N\to +\infty$ is exactly the Gauss circle problem. In this case the first term of the asymptotic expansion is $\frac{\pi}{4}N$ for similar reasons.
The text passage discusses a mathematical problem involving a sum with a floor function and explores the possibility of finding a closed-form solution. It also mentions the relationship between the variables \( n \) and \( N \) and provides an integral approximation for the sum.
no_question

the passage does not contain a direct mathematical question that requires an answer. instead, it presents a problem and discusses various aspects and approximations related to it.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Closed-form for Floor Sum 3 - With knowledge of inner expression

Consider the following sum: $$\sum_{k=0}^{n}\left\lfloor\sqrt{k^2+N}\right\rfloor$$

Assume we know the factorization of $N$, in other words, we know for which $k$, $k^2+N$ will be square, according to Fermat's Factorization Method.

$N$ in this case is a positive odd integer.

Can we use this knowledge to get a closed-form for the sum?

Thanks a lot

How large is $n$ compared to $N$? Is $n\approx\sqrt{N}$? You are counting the lattice points in a region bounded by a hyperbola, and $\sqrt{x^2+N}$ behaves like $x$ for large values of $x$. Additionally, it is not likely that your expression has a nice closed form, but it is for sure pretty close to $$\int_{0}^{n}\sqrt{x^2+N}\,dx = \frac{1}{4} \left(2 n \sqrt{n^2+N}-N \log N+2 N\log\left(n+\sqrt{n^2+N}\right)\right)$$ which if $n=\sqrt{N}$ behaves like $\left(\sqrt{2}+\log(1+\sqrt{2})\right)\frac{N}{2}\approx\frac{7}{6}N$ for large values of $N$.
Also notice that the problem of finding the first two terms of the asymptotic expansion of $$\sum_{k=0}^{\sqrt{N}}\left\lfloor\sqrt{N\color{red}{-}k^2}\right\rfloor$$ as $N\to +\infty$ is exactly the Gauss circle problem. In this case the first term of the asymptotic expansion is $\frac{\pi}{4}N$ for similar reasons.
user
The text passage discusses a mathematical problem involving a sum with a floor function and explores the possibility of finding a closed-form solution. It also mentions the relationship between the variables \( n \) and \( N \) and provides an integral approximation for the sum.

TYPE: NO_QUESTION

The passage does not contain a direct mathematical question that requires an answer. Instead, it presents a problem and discusses various aspects and approximations related to it.