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

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1answer
42 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 $0$.
3
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1answer
58 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
161 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
39 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
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4answers
342 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
97 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
95 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
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1answer
23 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
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3answers
64 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
60 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
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0answers
40 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?
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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
23 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
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0answers
96 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
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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
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0answers
14 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
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1answer
40 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
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1answer
76 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
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0answers
65 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 ...
10
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1answer
145 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 ...
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0answers
72 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 ...
6
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0answers
75 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
319 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 ...
14
votes
1answer
156 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
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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
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1answer
47 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
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0answers
86 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
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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
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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 ...
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0answers
41 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
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1answer
57 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
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1answer
35 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
467 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
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1answer
40 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 ...
2
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0answers
68 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 $n$. ...
1
vote
2answers
36 views

Why is there a pattern for making orders of perfect squares (first one, second one, third one) by simply adding two to the next adding each time? [duplicate]

For example, if I had a perfect square of $16$, which is the fourth perfect square, I would add nine to get to the fifth perfect square, $25$. This is probably how it ...
2
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4answers
156 views

How can I solve $m^2+n^2=5077$

$\forall (m,n)\in\mathbb{Z}$, I'm looking for an efficient way to solve this equation $$ m^2+n^2=5077 $$
2
votes
1answer
70 views

Numbers which are not the sum of distinct squares

We are defining square factorization as representation a positive natural number as sum of squares of different positive, integer numbers. For example $5 = 1^2 +2^2$ and $5$ has no more ...
4
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0answers
66 views

Prove $\forall n\in\mathbb{N}, \exists m\in\mathbb{N}; n=\pm1^2\pm2^2\pm\cdots\pm m^2.$

And we choose the positive and negative signs in a way that the equation becomes true. I think it can be proved with mathematical induction. So here's how I begin: For $n=1$, $1=+1^2$ which is true. ...
6
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1answer
129 views

Numbers as sum of distinct squares

Yesterday Polish Olympiad of Information Science ended, one of the questions was purely mathematical, Squares (PL). In the task, we have defined square ...
2
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2answers
57 views

Are there no even squares expressible as the sum of two prime squares?

When I was playing around with different number sequences, I noticed that I couldn't find any even squares that are expressible as the sum of two prime squares. Is this true, and is this related to ...
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0answers
64 views

When $Ax^2+By^2=z^2$ has a solution in integers?

Consider the Diophantine equation $Ax^2+By^2=z^2$, with positive integer parameters $A$ and $B$ (not necessarily square-free or co-prime). When does this equation have a non-trivial solution? Can one ...
0
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1answer
44 views

how can i calculate square sum of all the products of the summation?

Rule.  The square of a sum is equal to the sum of the squares of all the summands plus the sum of all the double products of the summands in twos: But I do not know how to calculate the square of ...
4
votes
2answers
70 views

If $(a^2+b^2) \mid (c^2+d^2)$ and $\gcd(a,b)=\gcd(c,d)=1$ and $\gcd(a,c)>1$, what can be said about the components?

While working on a divisibility problem in integers $a,b,c,d$, with $\gcd(a,b)=\gcd(c,d)=1$, I've come up against the hypothetical condition $$ (a^2+b^2) \mid (c^2+d^2), \tag{$\star$} $$ where, also ...
2
votes
1answer
67 views

sum of square root of primes 2

I dont know how to solve the problem below. (1) $p[1]$, $p[2]$, $\ldots$, $p[n]$ are distinct primes, where $n = 1,2,\ldots$ Let $a[n]$ be the sum of square root of those primes, that is, $a[n] = ...
2
votes
3answers
104 views

Sum of squares of two integers divisible by five [closed]

Supposing $x,y$ are natural numbers, what is the probability that the sum of their squares are divisible by 5? I am getting $1/3$ as squares can only end with $0,1,4,5,6,9$. So $36$ pairs are ...
0
votes
1answer
60 views

Number of integer lattice points within a circle

I am trying to solve a problem on codeforces, to be precised, this problem. I was able to figure out that the solution is $N(n) - N(n-1)$ where $N(n)$ is the number of lattice points withing a circle ...
1
vote
1answer
75 views

In how many ways can a number be expressed as a sum of squares of two natural numbers? [duplicate]

In how many ways can $145^2$ be expressed as sum of two squares? I tried solving it by finding out the Pythagoren triplets. $145= m^2+n^2 = 12^2+1^2$ & $9^2+8^2$ so triplet is $(145, m^2-n^2 , ...
0
votes
2answers
79 views

Represent a prime number $p$ congruent to $1$ $\pmod{3}$ by a sum of a square and $3$ times a square

I want to have a proof of the fact that each prime number is the sum of a square and three times a square (Euler). Context I read the answer to my former question about the number of points on an ...