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

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7
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80 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
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0answers
45 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
votes
0answers
71 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
286 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 ...
13
votes
1answer
140 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
26 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
43 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
77 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
20 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
57 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
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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
54 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
15 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
450 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
33 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
28 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
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2answers
31 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 ...
1
vote
1answer
53 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
58 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. ...
5
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1answer
96 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
votes
2answers
49 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
60 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
32 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
65 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
60 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] = ...
0
votes
0answers
29 views

Reducing a sum of four squares to a sum with root sum equal to $1$

It is well-known that every odd natural number can be written as the sum of four squares. Perhaps less well-known is the fact that every odd natural number can be written as the sum of the squares of ...
2
votes
3answers
88 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
40 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
64 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
75 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 ...
2
votes
0answers
34 views

Polynomial Density Theorem

So I was consider Lagrange's 4-square theorem and came up with this generalization: Given a polynomial with rational coefficients $$P(x) = a_0 + a_1x + ... + a_nx^n$$ Determine if there exists ...
0
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0answers
254 views

Prove the inequality $\sum_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $

A is a square matrix with positive elements and x is a real vector (both of them n>1 dimensional). Prove that for any such matrix and vector $$\sum\limits_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} ...
0
votes
1answer
30 views

Largest $k$ such that $(p-k) = \lceil \sqrt{(p-2k) p} \rceil$

Assume $p \in \mathbb P.$ Assume $0<p-2k<p$ and the next square larger than $p(p-2k)$ is $(p-k)^2$. It is trivial to show that $p(p-2k)+k^2$ is a square. Simply $p(p-2k)+k^2 = (p-k)^2.$ ...
0
votes
0answers
25 views

Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
1
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1answer
29 views

Maximum length of a representation of a number as an alternating sum of squares

Define a function $$\mathscr R: \mathbb N \to \mathbb N, \ \ \mathscr R(n) = \lceil \sqrt{n} \rceil ^2 -n.$$ IE. the distance of $n$ to the next prefect square. Sequence A068527 on OEIS. If $\mathscr ...
0
votes
4answers
200 views

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 ...
6
votes
4answers
203 views

When is the sum of two squares the sum of two cubes

When does $a^2+b^2 = c^3 +d^3$ for all integer values $(a, b, c, d) \ge 0$. I believe this only happens when: $a^2 = c^3 = e^6$ and $b^2 = d^3 = f^6$. With the following exception: $1^3+2^3 = 3^2 + ...
2
votes
2answers
199 views

Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares.

I have figured out a (long, and tedious) way to do it. But I was wondering if there is some sort of direct correlation or another path that I completely missed. My attempt at the program was as ...
0
votes
1answer
39 views

Find $2^{2}+4^{2}….(2n)^{2}$?

I tried subtracting $1^{2}+3^{2}….n^{2}$ from $2^{2}+4^{2}….2n^{2}$, but that didn't work. I know the answer is
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vote
3answers
84 views

integer solutions to $x^2+y^2+z^2+t^2 = w^2$

Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?
0
votes
2answers
162 views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
0
votes
2answers
26 views

Non negativity condition for quartic polynomials?

Say I have a quartic polynomial $f(x) = ax^4 + bx^3 + cx^2 + d$. I am told that $f(x)$ is nonnegative iff it can be expressed as a sum of squares as follows. $f(x) = \sum_{i=1}^4 q_i(x)^2$. As an ...
7
votes
3answers
308 views

If a prime can be expressed as sum of square of two integers, then prove that the representation is unique.

If a prime can be expressed as sum of two squares, then prove that the representation is unique. My attempt: If $a^2+b^2=p$, then it is obvious that $a,b$ of different parity. Now, I assume the ...
0
votes
2answers
59 views

Problem Solving Question? Sum of the squares

The sum of the squares of two numbers is 247 and the product of the two numbers is 21. How would I find all possible values for the sum of the two numbers?
0
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3answers
150 views

$2017$ as the sum of two squares

Write the prime $2017$ as the sum of two squares $2017$ can be written as the sum of two squares because it is a prime of the form $p\equiv 1\ ($mod $4)$ Using an appropriate algorithm find the two ...
4
votes
2answers
112 views

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$.

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$. Here $n\in\mathbb N$, $a,b,c,x,y,z\in\mathbb Z$. This problem is originally ...
4
votes
1answer
106 views

Fibonacci numbers expressed as squares of lower Fibonacci numbers

I am no mathematician so my apologies for my ignorance. I notice that every number in the Fibonacci series can be expressed as a previous Fibonacci number squared plus or minus (alternating) another ...
2
votes
3answers
99 views

Does this equation have positive integer solutions?

The only solution I can find for $a^2 + b^2 + c^2 = d^2 + e^2 + f^2$ is $a=0$, $b=0$, $c=0$, $d=0$, $e=0$, $f=0$. Are there any positive integer solutions? Any where none of $a,b,c,d,e,f$ are ...
0
votes
1answer
43 views

Difference of squares - number of representations

There exists a well-known result concerning a number of representations of $n$ as a sum of two squares. Is there anything similar for a number of representations of $n$ as a difference of two squares? ...
3
votes
1answer
69 views

Azuma's inequality: Expected sum of differences

I am looking for an extension of Azuma's inequality which involves the expected sum of squared differences. In particular, recall that Azuma's inequality states \begin{align*} \Pr[X_n-X_0 \geq a] \leq ...