text: # 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.
explanation: The text extract discusses a mathematical problem involving a sum with a floor function and the square root of a quadratic expression. The main question is whether a closed-form expression can be derived for the sum given the factorization of \( N \). The extract also includes additional commentary on the behavior of the sum and its relation to the Gauss circle problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract discusses a mathematical problem involving a sum with a floor function and the square root of a quadratic expression. The main question is whether a closed-form expression can be derived for the sum given the factorization of \( N \). The extract also includes additional commentary on the behavior of the sum and its relation to the Gauss circle problem.
  
  Label: question-and-answer