### nt.number theory - Question on Sums of Squares

Is it possible for two different $n$-element sets, each of which consists of $n$ unique positive integers (they can appear in both sets, though) to have the same sum when the squares of their elements are added?Edit: For obvious reasons, I'm not considering the case $n=1$....Read more

### quadratic residues - Sum of squares modulo a prime

What is the probability that the sum of squares of n randomly chosen numbers from $Z_p$ is a quadratic residue mod p?That is, let $a_1$,..$a_n$ be chosen at random. Then how often is $\Sigma_i a^2_i$ a quadratic residue?...Read more

### Can two the sum of two primitive perfect squares be equal? $a^2+b^2 = c^2 + d^2$?

Given a primitive perfect square, $n^2=a^2+b^2$ where $gcd(a, b) = 1$ and$m^2=c^2+d^2$ where $gcd(c,d)=1$Can $n=m$? a, b, c, d, n, and m are positive integers. And I forgot to mention:EDIT 1: Two of (n, a, b) are divisible by 3. One of (n, a, b) is divisible by 9. And the same for m, c, and d:Two of (m, c, d) are divisible by 3. One of (m, c, d) is divisible by 9.I have not found a situation among the first few hundred lowest perfect squares where $n=m$ and I would like to make sure this is true.I know that any square can be represented by a su...Read more

### How to show that sum of squares is n times of mean?

I gave $X_i\sim N(\mu=8, \sigma^2=1)$ for $i=1,...91$ with observed $\bar{x}=7.319$ and I calculate $f(\bar{x}=7.319|\mu_0=8)$ and I stuck in one step of calculation, actually the very first: $(\frac{1}{\sigma\sqrt{2\pi}})^n \exp(-\frac{\sum (x_i-\mu_0)^2}{2\sigma^2})= (\frac{1}{\sigma\sqrt{2\pi}})^n \exp(-\frac{n (\bar{x}-\mu_0)^2}{2\sigma^2})$How come that:$\sum(x_i - \mu_0)^2 =n(\bar{x}-\mu_0)^2$?Please, help, I've been thinking and calculating for hours and still I can't reach the right side from the left....Read more

### Represent an integer as a sum of n non-consecutive squares

Is it possible to decompose an integer into a sum of n unique squares ,even though they are not necessarily consecutive.For instance, How would I obtain the sequence 1*1 + 7*7 + 14*14 + 37*37 given the integer 1615or 11*11 + 15*15 + 29*29 + 43*43 + 69*69 given the integer 7797....Read more

### sums of squares - Is there a pattern defining the existence of root integer distances in an isometric grid?

In a standard square grid pattern the distances to integer root locations is simply the sum of two squares. We find that these distances have $\sqrt1, \sqrt2, \sqrt4, \sqrt5, \sqrt8, \sqrt9, \sqrt10, \sqrt13, ...$ But, knowing it's the sum of two squares we know this is Fermat's theorem on the point and that these exist so long as the prime factors of the number are even for $4k + 3$. So $3$ is absent because $3$ has prime factors of $3$ and there's 1 $3$ which is odd and $4*0+3 = 3$. $6$ is missing as it has prime factors $2,3$, which is again...Read more

### Sums of squares minus square of sums

I have the following equation in a statistics textbook and cannot see how the right side comes into being.$$\frac{1}{n} \sum_{i=1}^n x^2_i - \left(\frac{1}{n} \sum_{i=1}^n x_i\right)^2= \frac{1}{n} \sum_{i=1}^n (x_i-\bar{x_n})^2$$...Read more

### $5^m$, where m is any natural, can be expressed as the sum of two perfect squares?

Prove that for all natural $m$, $5^m$ can be expressed as the sum of two perfect squares.Also, prove that $5^m + 2$ can be expressed as the sum of three perfect squares....Read more

### Sum of square roots...........

$$\mbox{If}\quad S =1 + \,\sqrt{\,\frac{1}{2}\,}\, + \,\sqrt{\,\frac{1}{3}\,}\, +\,\sqrt{\,\frac{1}{4}\,}\, +\,\sqrt{\,\frac{1}{5}\,}\, + \cdots + \,\sqrt{\,\frac{1}{100}\,}\,\,,$$then what is the value of $\left\lfloor\,S\,\right\rfloor$ ?.Here $\left\lfloor\,S\,\right\rfloor$ is the greatest integer function which is less than or equal to $S$....Read more

### Upper bound on sum of square of integers

I have $n$ non-negative integers $x_1, \dotsc, x_n$ which satisfy the constraint $\sum x_i = S$I want to derive a bound on $\sum x_i^2$. An easy bound can be calculated as: $\sum x_i^2 \le (max_{x_i}) \sum x_i = S^2$ This bound works for non-negative reals. Is there a tighter bound for non-negative integers or is this the best we can do?...Read more

### nt.number theory - Sums of Squares

Every prime $p = 4k + 1$ can be uniquely expressed as sum of two squares, but for whichintegers $x$ is $x^2 + y^2 =$ some prime $p$? Stated differently, does the square ofevery positive integer appear as one of the squares in the representation of someprime $p$?...Read more