# Tagged Questions

For questions concerning various representation of integers as sums of squares, which are studied in number theory.

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### Is $2\bar x(1 - \bar x) - \sum_{i=1}^n 2 x_i (1-x_i) = 2 \sum_{i=1}^n (x_i - \bar x)^2$ true?

In Nei 1973, right after equation (9), the author says: [..] it can be shown that $H_T = 2\bar x(1 - \bar x)$ and $D_{st} = 2 \sigma_x^2$, where $\bar x$ and $\sigma_x^2$ are the mean and variance ...
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### Showing Residual Sum of Squares for Multiple Linear Regression is 0

Problem: I have the linear regression model: $y_i=\beta_0+\sum_{k=1}^p \beta_kx_{ik}+\epsilon_i$ where $\epsilon_i\sim N(0,\sigma^2)$, for $i = 1,2,\ldots ,n$. I want to prove that the residual sum ...
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### prove if n - natural number divide number $34x^2-42xy+13y^2$ then n is sum of two square number

prove if n - natural number divide number $34x^2-42xy+13y^2$ where x,y are relatively prime then n is sum of two square number. I don't know what is going on in this exercise. I will be grateful ...
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### Additive basis of order n: Sets which allow every integer to be expressed as the sum of at most n members of that set. [closed]

Every integer can be expressed as the sum of at most 3 triangular numbers. That is, the set of triangular numbers is an additive basis of order 3. The sum of the inverse triangular numbers is 2. (1/1 +...
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### Show that if $n\equiv 3, 6 \pmod9$ then $n$ is not a sum of two squares

Show that if $n\equiv 3, 6 \pmod9$ then $n$ is not a sum of two squares. I started by: Assume $n=a^2+b^2$ a sum of two squares. Then $a^2,b^2\equiv 0,1,4,7 \pmod9$, and no combination these numbers ...
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### Student test statistic and self normalizing sum.

I study the asymptotic distribution of self normalizing sums which are defined as $S_n/V_n$ where $S_n=\sum_{i=1}^n X_i$ and $V_n^2 = \sum_{i=1}^n X_i^2$ for some i.i.d RV's $X_i$. Motivation ...
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### Is there a theory of “sums-of-squares residues”?

The theory of quadratic residues is long- and well-studied. Recall that, [somewhat simplified] if $x,a,b$ are integers, with $0 \le a < b$, such that $$x \equiv a^2\!\!\!\pmod{b},$$ then we say ...
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### Probability of having at least $j$ collisions when tossing $m$ balls into $n$ bins

Suppose that we throw $m$ balls into $n$ bins uniformly and independantly at random. We consider collisions as distinct unordered pairs, e.g., if 3 balls are tossed in one bin, we count 3 collisions. ...
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### Triangular Numbers and Sum of Two Squares

"If n is a triangular number, show that each of the three consecutive integers, $8n^2, 8n^2+1, 8n^2+2$ can be written as a sum of two squares." I have spend hours working on this problem and cannot ...
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### Maximize sum of squares

Lets say that I know that $n$ values $x_i$ sums up to $\mu$: $$\mu=\sum_{i=1}^n x_i$$ I also now that $0\leq x_i\leq 1$ for all $i=1\cdots n$. I want to find an upper bound as tight as possible ...
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### Why is the square root of a sum not equal to the square root of each its addends?

Example: Let's presume one was attempting to isolate m below: A common mistake would be: $k^2 = m^2 + n^2 \to k = m +n$ Even though: $k^2 = m^2 + n^2 \to k \neq m +n$ If you apply a square root to ...
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### What is the relation between the square root of the sum of squares and the sum of the absolute values?

I want to prove that $\sqrt{\sum a_{i}^{2}} \geq \sum \left | a_{i} \right |$, is it possible ?