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

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3
votes
2answers
85 views

Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
0
votes
0answers
34 views

Bounded pythagorean triples

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
1
vote
0answers
23 views

From 4 squares to 2 squares

Let $r_k(n)$ denote the number of representations of $n$ as a sum of $k$ squares of integers. Suppose I know that $r_4(n) = 8\sum_{4 \nmid d \mid n} d$ (Jacobi's Theorem). Is there a way to deduce ...
1
vote
0answers
58 views

Integer $2n^2+2$ as the sum of 2,3,4, and 5 squares

If $n-1$ and $n+1$ are both primes, establish that the integer $2n^2+2$ can be represented as the sum of 2, 3, 4, and 5 squares. I managed to solve 2 and 4 squares, since: $$2n^2+2 = ...
1
vote
0answers
44 views

On Catalan's complete solution to the equation $T^2=U^2+V^2+W^2$

Catalan proved the following: If $t,u,v,w$ are coprime integers such that \begin{equation*} t^2 = u^2 + v^2 + w^2, \end{equation*} then there exist integers $\alpha,\beta,\gamma,\delta$ such that ...
0
votes
0answers
31 views

Square matrix whose sum of squared elements equals 1.

I'm doing some applied work where I've come across examples that involve real valued square matrices that hold the following property, which expressed using tensor notation is $$A_{ij}A_{ij} = 1$$ ...
4
votes
4answers
66 views

Express 2104 as the sum of four squares

How to write 2104 as the sum of four squares. I know the general equation for factoring a number into the sum of four squares but I don't know how to go about this when some of the prime factors are ...
3
votes
2answers
71 views

How to show that that the following three consecutive numbers $3^{2^{10}} − 1$, $3^{2^{10}}$,$3^{2^{10}}+ 1$ are the sum of two squares?

Show that the following three consecutive numbers: $$ 3^{2^{10}} − 1, 3^{2^{10}} , 3^{2^{10}} + 1 $$ can be represented as sums of two integer squares.
0
votes
5answers
76 views

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

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.
1
vote
2answers
95 views

How to derive the formula for the sum of the first $n$ perfect squares? [duplicate]

How do you derive the formula for the sum of the series when $S_n = \sum_{j=1}^n j^2$? The relationship $(n, S_n)$ can be determined by a polynomial in $n$. You are supposed to use finite differences ...
1
vote
0answers
43 views

Combining sums and/or differences of squares

I'd like to combine a sum of as many squares as possible into a sum of as few squares as possible. The signs of the squares doesn't matter. For example, the Brahmagupta-Fibonacci Identity combines a ...
1
vote
1answer
52 views

Find all natural solutions to $x^2+2y^2 = z^2$ [duplicate]

I need to find all natural solutions to $x^2 + 2y^2 = z^2$ What I tried: I did $\pmod 2$ to the equation receiving $z^2 - x^2 \equiv 0 \pmod 2$. Then there are two possibilities: $x^2 \equiv 0 ...
2
votes
6answers
66 views

find all natural solution that satisfy $x^2+y^2 = 3z^2$

I need to find all natural $x,y,z$ that satisfy the following $x^2+y^2 = 3z^2$ $(0,0,0)$ is an answer of course. What I tried: I tried solving with congruences. I know that every square number ...
5
votes
3answers
70 views

Is it possible to have $a^2 + b^2 = c^2 + 1$ for $a$, $b$, $c$ being coprime integers?

As stated above. I'm working on a possible proof. It appears that $$(b+1)(b-1)=(c+a)(c-a)$$ That's where I'm stuck. Any help please? A clear, simple proof desired, thanks!
1
vote
1answer
45 views

Show that there are infinitely many primes $p$ of the form $p=a^2+b^2+c^2+1$

I know that any prime can be written as the sum of four squares. But I don't know how to know one of these squares is $0$.
5
votes
1answer
96 views

Is it known whether any positive integer can be written as the sum of $n$ different squares?

Is it known whether any sufficiently large positive integer can be written as the sum of four different squares? I know that every positive integer can be written as the sum of four not necessarily ...
-3
votes
2answers
222 views

square of digits - why does it always contain 1 or 89 [closed]

I attempted project euler problem 92, while I passed it, my solution works, but had just...awful performance. So I would like to try again tomorrow. In the meantime understanding why the iteration ...
1
vote
2answers
43 views

Expressing a metric as a sum of (possibly) many squares

Given a Riemannian manifold $M$ whose metric $g$ has zero curvature, it is known that we can find local coordinates $x^i$ such that $$g=\sum_{i=1}^{\dim(M)}(dx^i)^2.$$ Conversely, if the curvature ...
2
votes
4answers
416 views

Sum of square patterns

Can anyone give the name of this pattern $$136^2+137^2+138^2+139^2+140^2+141^2+142^2+143^2+144^2 =\\ 145^2+146^2+147^2+148^2+149^2+150^2+151^2+152^2$$
0
votes
2answers
186 views

How can I add 2 squares geometrically to get a bigger square?

Suppose all I have is 2 square pieces of paper of equal size and a pair of scissors. How can I cut the paper and rearrange the pieces into 1 bigger square (combined size)? I presume it would include ...
3
votes
3answers
99 views

Find rational points on $x^2 + y^2 = 3$ and on $x^2 + y^2 = 17$

$(a)$ Find all rational points on the circle $x^2 + y^2 = 3$, if there are any. If there is none, prove so. $(b)$ Find all rational points on the circle $x^2 + y^2 = 17$, if there are any. If there ...
0
votes
1answer
27 views

Is there a system of equations model to solve sum of squared integers?

I have made a wooden block model (5x5,4x4...1x1) to model geometrically "sum of squared integers" for my students to examine. aside from using plug-n-play with the standard formula given from our ...
1
vote
3answers
116 views

The number of ways of writing an integer as a sum of two squares

Given an integer $m=pq$, where $p,q$ are both primes such that $p\equiv 1 \pmod{4}, q\equiv 1 \pmod{4}$. It is known that $p$ can be written as a sum of two squares (of positive integers) in a unique ...
2
votes
1answer
70 views

writing $pq$ as a sum of squares for primes $p,q$

Let $p$ and $q$ be distinct primes congruent to $1$ mod $4$. How many ways are there to write $pq$ as a sum of squares? I know that any prime $p\equiv 1\pmod 4$ can be written uniquely as a sum of ...
2
votes
0answers
41 views

Number of ways to write a prime $p \equiv 1 \bmod 4$ as a sum of two squares [duplicate]

My intuition, from checking many cases, is that there is only ONE way to write a prime $p \equiv 1 \bmod 4$ as a sum of two squares. But how can I formally show this?
1
vote
2answers
40 views

What does it mean for an integer $n$ to have $k$ expressions as $x^2 + y^2$?

I need to write a Maple procedure to find the first integer $n$ with $k$ representations as $x^2 + y^2 = n$. I do not really understand what this statement means
1
vote
1answer
26 views

Why multivaraite positive polynomials cannot be written as sum of squares?

It is wellknown that a positive univariate polynomial $p(x)>0$ for all $x\in R$, can be written as a sum of squares: $p(x) = \sum_{i=1}^n q_i^2(x)$, and I found references saying (without any ...
7
votes
0answers
102 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
2
votes
0answers
22 views

Nonnegative vs SOS

Consider the polynomial $f(x_1, \cdots, x_n)$, I want to characterize $f$ being nonnegative, i.e., $f\geq 0$. For $n=1$, this is equivalent to saying that $f$ is SOS (sum of square). However, in ...
1
vote
0answers
17 views

Mutual difference of vectors squared, does it have a name?

Given a set of $n$ vectors $\def\vv{\vec{v}} \vv_i$ with the additional property that they all have the same absolute value $||\vv_i||=c$, define the average of the vectors as $\vv = ...
0
votes
1answer
43 views

Can sum of 3 composite squares be a perfect square?

Do $3$ composite natural numbers exist - $X$, $Y$ and $Z$, such that $X^2+Y^2+Z^2$ is a perfect square? If yes, please give an example, I need it for another proof.
0
votes
1answer
36 views

Let $F$ be a field and let $a,b,c,d\in F\Rightarrow (a^2+b^2)(c^2+d^2)$ can be written as $x^2+y^2$ for some $x,y\in F$?

Let $F$ be a field and let $G_2$ be the subset of $F$ consisting of all elements which can be written as a sum of $2$ squares of elements of $F$. Is the product of two elements of $G_2$ again an ...
4
votes
1answer
79 views

Let $F$ be a field in which we have elements satisfying $a^2+b^2+c^2 = −1$. Show that there exist elements satisfying $d^2+e^2 = −1$.

Let $F$ be a field in which we have elements $a, b$, and $c$ satisfying $a^2+b^2+c^2 = −1$. Show that there exist elements $d$ and $e$ of $F$, satisfying $d^2+e^2 = −1$. Any hint? This is an ...
0
votes
0answers
69 views

Reduce $(a+b)^2(c+d)^2-16abcd$ to sum of squares

Is it possible to reduce $$(a+b)^2(c+d)^2-16abcd$$ to sum of squares? This expression is used in proof of AM-GM inequality. It is known that $$\dfrac{a+b}{2}\ge\sqrt{ab}\tag{1}$$ So, it can be proved ...
13
votes
1answer
195 views

Integer solutions to $\prod\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}x_i^2$

Given integers $x_1,\dots,x_n>1$. Let's assume WLOG that ${x_1}\leq\ldots\leq{x_n}$. I want to prove that the only integer solutions to any equation of this type are: $x_{1,2,3 ...
1
vote
0answers
89 views

Sum of two squares - Number of steps in Fermat descent

If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv -1 \mod p$. Is there a possibility to say ...
7
votes
0answers
80 views

Applying iterated function on the sum of the squares of the prime factors of $30$

Let $f(n)$ denote the sum of the squares of the prime factors of $n$ with multiplicity. For example, $f(60)=f(2\cdot2\cdot3\cdot5)=2^2+2^2+3^2+5^2=42$. Denote the iterated function ...
3
votes
3answers
338 views

Pythagorean triples

So I am given that $65 = 1^2 + 8^2 = 7^2 + 4^2$ , how can I use this observation to find two Pythagorean triangles with hypotenuse of 65. I know that I need to find integers $a$ and $b$ such that ...
17
votes
1answer
216 views

Numbers that are the sum of the squares of their prime factors

A number which is equal to the sum of the squares of its prime factors with multiplicity: $16=2^2+2^2+2^2+2^2$ $27=3^2+3^2+3^2$ Are these the only two such numbers to exist? There has to be an ...
1
vote
1answer
29 views

Prove by induction that $\sum_{k=0}^{n}(-1)^{n+k} k^{2} = \frac{n(n+1)}{2}$

Prove by mathematical induction that $\forall n \in \mathbb{N}:~~~~ \sum_{k=0}^{n}(-1)^{n+k} k^{2} = \frac{n(n+1)}{2}$ Step 1: Show true for $n = 1$: LHS: $(-1)^{(1+0)} \cdot 0^{2} + (-1)^{(1+1)} ...
2
votes
1answer
49 views

Suppose $f:N \to N$ is increasing. Does there exist an $M$ such that $m=\sum_{i=1}^M f(a_i)$ always has a solution?

There are a lot of number theory problems like the following: (1) Find all $m$ such that $m=a^2+b^2$, or all $m$ such that $m=a^2+b^2+c^2$, or even $m=a^2+b^2+c^2+d^2$. (2) Find all $m$ such that ...
1
vote
0answers
88 views

SOS relaxations for polynomial optimization

I do not understand how SOS (Sum-Of-Squares) relaxation for polynomial optimization works in some cases. For instance, consider the polynomial optimization problem: \begin{equation} ...
2
votes
1answer
21 views

Clarification on how to prove polynomial representations exist for infinite series

With reference to this question, I would like a clarification of the comment given by @Ant (but someone else could answer instead). I basically have 2 questions: Is there any formal way to prove ...
0
votes
1answer
59 views

A function f(n), that for any n gives the sum of 1^2 up to n^2?

I was solving Project Euler's problem # 6, where one has to find the sum of square of numbers from 1 to 100. I solved it using code and downloaded the Overview provided, to learn from it but I can't ...
1
vote
0answers
47 views

Given a closed form for a series, what can be said about the sum of the squares of its terms?

Suppose I have an infinite integer sequence $\{a_k\}$, and suppose I know a closed form in terms of $n$ for this sum: $$\displaystyle\sum\limits_{k=1}^{n} a_k$$ Given this, is it always (or ever) ...
1
vote
1answer
58 views

primes of the form $4k+3$ and sums of squares [duplicate]

It is well-known that if $p$ is a prime of the form $4k+3$ and $p|x^2+y^2$ then $p|x$ and $p|y$. I forget what is the name of this result, and where can I find a proof (please provide a link).
1
vote
1answer
50 views

Lagrange Method for Presenting Bilinear form as sum of squares

I have the following question in my assignment which I'm having a hard time solving. For the following bilinear form, present find a digonal form (diagonal matrix form): What I thought to do at ...
11
votes
2answers
474 views

(Non?)-uniqueness of sums of squares

(I've had almost no exposure to number theory, so please keep answers as elementary as possible.) Write $\mathbb{N} = \{0,1,2,3,\ldots\}$ for the natural numbers. Then every element of $\mathbb{N}$ ...
1
vote
1answer
46 views

Application of the Jacobian

I have been stuck on this question for a while now to no success. Help would be appreciated. Consider $x$ and $y$ such that $(x, p) =(y, p) = 1$. For what $p$ does their exist $x$ and $y$ such ...
3
votes
0answers
95 views

Which integers are a sum of two relatively prime squares?

It's well known that a positive integer $n$ is a sum of two squares if and only if every prime of the form $4m + 3$ that divides $n$ appears with even multiplicity in the prime factorization of $n$. ...