Combinatorics: How Many Vectors with Square Sum = K?

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The discussion centers on determining the number of combinations of natural number vectors whose squared components sum to a given natural number K. Participants explore mathematical approaches and the implications of the problem, referencing the Sum of Squares Function for insights. The complexity of the problem is acknowledged, with users expressing frustration over its difficulty. There is a call for clarity on the combinatorial aspects and potential solutions. Overall, the conversation highlights the challenge of solving this combinatorial problem effectively.
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Given a Natural number K, how many combinations \[<br /> x = \left( {x_1 ,...,x_N } \right)<br /> \]<br /> of Natural numbers vectors are there, so that \[<br /> \sum\limits_{i = 1}^N {x_i ^2 } = K<br /> \]<br />?

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Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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