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

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2
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0answers
29 views

What came first: pythagoras number or pythagorean fields? [migrated]

Which concept was first introduced: the pythagoras number of a field or pythagorean fields? I have not found anything on this matter, but my gut feeling says the latter. One can more directly link the ...
1
vote
3answers
62 views

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 ...
4
votes
2answers
48 views

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 ...
3
votes
0answers
15 views

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 ...
4
votes
0answers
81 views

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. ...
1
vote
3answers
42 views

Solving $rX_1^2+sY_1^2+tZ_1^2=rX_2^2+sY_2^2+tZ_2^2$ completely in integers

Given pairwise relatively prime integers $r,s,t$, I’m looking for a complete solution (i.e., integer parameterization or similar) for the Diophantine equation $$ ...
0
votes
2answers
28 views

Pattern in digits of sums of consecutive squares

I am interested in patterns in square numbers as well as the reasons behind them and I can't seem to figure out (also how to prove) why do the sums of two consecutive squares only end in digits 1, 3 ...
1
vote
1answer
56 views

Prove that $4n+2=x^2+y^2+z^2$ for some odd $x,y$ and even $z$

Show that for all $n\in \mathbb{N}$, exists $x,y,z \in \mathbb{N}$, such that $x,y$ are odd and $z$ is even, such that $4n+2=x^2+y^2+z^2$. I started by using the fact that every natural number has a ...
2
votes
0answers
67 views

$m^2+n^2$ and $m^2-n^2$ cannot both be squares [duplicate]

I need to show that there aren't any $m$ and $n$ such that $m^2+n^2$ and $m^2-n^2$ are both squares. First of all, assume without loss of generality that $m$ and $n$ are co-prime, since otherwise we ...
2
votes
3answers
85 views

Show that $\forall n\in\mathbb{N}$, $14^n$ can be represented as a sum of three perfect squares.

Show that $\forall n\in\mathbb{N}$, $14^n$ can be represented as a sum of three perfect squares. I checked $(\mod 7)$ and deduced that the three squares can be $1,4,2(\mod 7) $ or all divisible by ...
0
votes
3answers
34 views

What does the algorithm s = s + k * k do?

I just finished an exam in my math class and I did well except for one question that I just can't get out of my head, it seems simple but I just can't figure it out: PROBLEM: ...
4
votes
2answers
278 views

Sum of squares of integers divisible by 3

Suppose that $n$ is a sum of squares of three integers divisible by $3$. Prove that it is also a sum of squares of three integers not divisible by $3$. From the condition, ...
0
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0answers
28 views

Algorithm Identification

Background I'm currently working with a system that has a 4-dimensional function. Currently, an algorithm is used to speed up calculation of the final value via interpolation, and two of the ...
6
votes
3answers
131 views

My formula for sum of consecutive squares series?

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers. For exmaple, this first number in the ...
4
votes
2answers
65 views

Why is a polynomial $f(x)$ sum of squares if $f(x)>0 $ for all real values of $x$?

If a polynomial $f(x)>0 $ for all real values of $x$, then $f(x)$ is sum of squares. Why is this true ? I understand that the roots of this $f(x)$ will be complex and hence will exist as ...
4
votes
1answer
123 views

For which polynomials $f$ is the subset {$f(x):x∈ℤ$} of $ℤ$ closed under multiplication?

You surely know about the Brahmagupta–Fibonacci identity, $$(a_1^2 + b_1^2)(a_2^2 + b_2^2) = (a_1a_2 \pm b_1b_2)^2 + (a_1b_2 \mp a_2b_1)^2$$ which tells us that the product of two numbers, each of ...
2
votes
1answer
50 views

variant of Lagrange's four square theorem using a restricted set of squares

The well-known four square theorem states that any positive integer is the sum of at most four squares. Suppose that, instead of allowing all squares, I only consider the following set of squares: $$ ...
2
votes
4answers
74 views

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 ...
1
vote
1answer
23 views

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 ...
0
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1answer
32 views

Upper bound for sum of squares given mean of elements

I need upper and lower bounds as tight as possible for the following expression of the elements of a $n$ x $n$ matrix: $$ ...
4
votes
2answers
92 views

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions.

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions. I am not able to find an idea on how to proceed with the above questions. I have found only the obvious ...
0
votes
3answers
80 views

If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.

If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$. I feel that the problem basically uses ...
0
votes
0answers
36 views

If $p=x^2+y^2$ is a prime number, then $\left( \frac{x+y}{p} \right) = \left( \frac{2}{x+y} \right) $

Let $p=x^2+y^2$ be a prime number. How to prove that $\left( \dfrac{x+y}{p} \right) = \left( \dfrac{2}{x+y} \right) $ (where $\left(\frac ab\right)$ denotes the Jacobi symbol)?
0
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1answer
33 views

Relatively Prime Cases of the Brahmagupta-Fibonacci Identity

The Brahmagupta-Fibonacci identity states that the set of the sum of two squares is closed under multiplication. $$(u^2 + v^2)(A^2 + B^2) = (uA \pm vb)^2 + (vA \mp uB)^2$$ This is easy to verify as ...
0
votes
2answers
37 views

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} ...
1
vote
5answers
72 views

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 ...
1
vote
1answer
42 views

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 ?
1
vote
1answer
46 views

Prove that there are arbitrarily long sequences of consecutive integers, none of which can be written as the sum of two perfect squares.

Prove that there are arbitrarily long sequences of consecutive integers, none of which can be written as the sum of two perfect squares. First few numbers are ...
2
votes
1answer
50 views

Let $n$ be a positive integer. Show that $n$ and $2n$ have the same number of representations as a sum of two squares of non-negative integers. [closed]

Let $n$ be a positive integer. Show that $n$ and $2n$ have the same number of representations as a sum of two squares of non-negative integers. Hint: $(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$
3
votes
1answer
62 views

Sum of Consecutive Perfect Squares

If $n\in\mathbb{Z}$ and $2n+1$ is a perfect square, then is it true that $n+1$ is a sum of two consecutive perfect squares?
3
votes
3answers
92 views

Write $x_n=22..244…45$ as sum of $2$ squares

I've recently came across this problem and, although I've spent time looking for a solution, I don't have any interesting ideas. Let the numbers $$x_1=25$$ $$x_2=2245$$ $$x_3=222445$$ and ...
0
votes
1answer
61 views

Prove that for any prime $p$ there exist natural numbers $a,b$ for which $ p$ divides $a^2+b^2+1$ [closed]

Prove that for each prime $p$ there exist natural numbers $a,b$ for which $p$ divides $a^2+b^2+1$
0
votes
1answer
18 views

Hurwitz's matrix equations

I have a question about the proof of Hurwitz's 1-2-4-8 theorem about the sum of squares. I have consulted Chapter 1 of Rajwade's "Squares" book, notes by Keith Conrad, and notes by Daniel Shapiro. ...
1
vote
1answer
38 views

Can a number of the form $n = {2^r}{b^2}$ always be represented as a sum of two squares, if $r \geq 1$ and $b$ is an odd composite?

Can a number of the form $n = {2^r}{b^2}$ always be represented as a sum of two squares, if $r \geq 1$ and $b$ is an odd composite? More generally: Can a number of the form $n = {2^r}{b^2}$ ...
0
votes
2answers
64 views

Fermat's eleventh $F_{11}$ represented as sums of two squares?

There's an unanswered quest in another forum, that makes me sleepless: Fermat's Elevnth Let $a,b \in \mathbb{N^{+}}$ and $a>b$ then, there are $k$-representations for $F_{11}=2^{2^{11}}+1$ as sum ...
1
vote
2answers
33 views

Non negativity condition for polynomials?

Let $p(x)=3ax^2+2bx+(1-a-b), \,\,\,\,0<x<1$ What are the conditions on a and b to $p(x)>0$ Could someone give me a hint ?
3
votes
4answers
61 views

If a prime $p>2$ is expressible as $p=a^2+b^2$, then $4\mid (p-1)$

Show that if $p\in\mathbb{P}, p>2$ is a sum of squares, i.e $p=a^2+b^2$, then $p\equiv 1\pmod{4}$. Have established that the remainder can't be either of $0,2$ for obvious reasons. So we must ...
1
vote
1answer
25 views

Pythagoras number of $\mathbb{F}_p$

For a commutative ring $A$ and $a \in A$, define the length of $a$ as $$ l(a) = \inf \lbrace n \in \mathbb{N} \mid \exists a_1, \ldots, a_n \in A : a = \sum_{i=1}^n a_i^2 \rbrace . $$ Let $\Sigma A^2$ ...
0
votes
1answer
19 views

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$, deduce a formula for the sum of the squares of the positive divisors of $a$. -The section we ...
7
votes
2answers
194 views

Chinese New Year Equation 2016

In the spirit of Chinese New Year, here's a problem to commemorate the year. $\color{black}{\text{Solve the following equation for positive integers $a$ and $b$:}}$ ...
0
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1answer
35 views

Manipulating a sum and showing it has an upper bound

Let $x_1, \ldots ,x_n$ and $y_1, \ldots , y_n$ for $n \in \mathbb{N}$ be complex numbers such that $$\left| \sum_{k=1}^n x_k y_k \right| \leq 1. $$ Further I know that $$ \sum_{k = 1}^n |x_k|^2 \leq ...
0
votes
0answers
26 views

What is the distribution of sum of the squares of k dependent standard normal random variables?

It is known that the sum of the squares of k independent standard normal random variables is chi-squared distributed, what happens if we look at the sum of the squares of k dependent normal variables? ...
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0answers
28 views

Generalizing Landau-Ramanujan Theorem for sum of three squares

Let $S_{3}(x)$ denote the number of positive integers not exceeding x which can be expressed as a sum of three squares. Can we find an asymptotic formula for $S_{3}(x)$, maybe using Landau-Ramanujan ...
0
votes
1answer
22 views

Sum implies null product. True? [closed]

Let $x, y, z \in \mathbb{C}$. Assuming that $x+y+z=x^2+y^2+z^2=0$, is it true that $x^2y^2+y^2z^2+z^2x^2=0$?
2
votes
1answer
46 views

The sum of the squares of two factors of a number is a perfect square

Is there any way to get two factors of a number whose sum of the squares is a perfect square. As an example $19354423920$ is a number. which has $4262$, $4541160$ as factors ($19354423920 = 4262 * ...
1
vote
0answers
55 views

When is sum of squares a perfect square? [duplicate]

Recall that $$\sum_{j=1}^nj^2=\frac{n(n+1)(2n+1)}{6}.$$ When is this quantity a perfect square? It appears that the only solutions are $n=0,1,24.$ By setting $x=12n+6$, the problem reduces to finding ...
0
votes
2answers
55 views

Given a number N how many pairs of numbers have square sum less than or equal to N?

Let's define $F(N)$ as the number of pairs of distinct positive integers $(A, B)$ such that $A^2 + B^2 \leq N$ If $N=5$ the only possible such pair is $(1, 2)$, for $N=10$ the pairs are two: $(1,2)$ ...
7
votes
2answers
54 views

Show $a^2 + b^2 + 1 \equiv 0 \mod p$ always has a solution if $p = 4k+3$

If $p = 4k+3$ is a prime number (so $p = 7,11,19$ but not $p = 5,13$ or $p =15$) then there are numbers $a,b$ such that: $$a^2 + b^2 + 1 \equiv 0 \mod p$$ For example $2^2 + 3^2 + 1 = 14 = 7 \times ...
0
votes
4answers
51 views

What is 100,000 as a sum of some number of distinct squares? [closed]

I cannot find the solution to this problem. It is part of a larger homework question but I can't go on until I solve this question.
3
votes
2answers
48 views

Dividing the squares $1^2,2^2,\ldots,54^2$ into three equal groups with the same total sum

Is it possible to divide the squares $1^2,2^2,\ldots,54^2$ into three groups, each of which contains $18$ squares, such that the sum of squares within each group is the same for all three groups?