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

learn more… | top users | synonyms

2
votes
0answers
28 views

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

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
12 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
442 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 $\mathbb{N}$ ...
0
votes
1answer
26 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) and (y, p) = 1. For what p does their exist x and y such that $x^2 + ...
2
votes
0answers
25 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
29 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
49 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 ...
5
votes
1answer
86 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
41 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 ...
1
vote
0answers
59 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
votes
1answer
29 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
62 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
54 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
27 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
78 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
38 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
61 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
74 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
32 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
votes
0answers
251 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
29 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
24 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
vote
1answer
24 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
194 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 ...
2
votes
2answers
171 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
38 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
1
vote
3answers
83 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
137 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
24 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
297 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
58 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
votes
3answers
147 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
111 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
101 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
96 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
40 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 ...
5
votes
2answers
111 views

squares which are not the sum of a square and twice a triangular number

I'm trying to determine conditions on integer squares which cannot be written as a square and twice a triangular [all numbers positive], i.e. integers $n \ge 1$ where there are no integers $a,b \ge 1$ ...
4
votes
4answers
423 views

sum of 4 squares

Is there any natural number $A$ which cannot be written as: $$A=W^2+X^2+Y^2+Z^2$$ where $W,X,Y,Z \in \mathbb N \cup 0$ I was considering the fact that $a^2+b^2 \not = 1 \mod 4$ and was attempting to ...
2
votes
2answers
94 views

Prove that no $n,m, 0<n<m$ exist such that $m^2 +mn+n^2$ is a square number

Prove or disprove the claim that there are integers $n,m, 0<n<m$ such that $m^2 +mn+n^2$ is a perfect square.
2
votes
0answers
40 views

A four-digit square is of the form $aabb$. What's $a^2+b^2$? [duplicate]

The 5772 Ulpaniada included the following question: Consider a four digit square number (a number which is the square of a whole number).Its digit notation is $aabb$ (the thousands digit is $a$, ...
7
votes
1answer
193 views

Algorithm to find solution to $ax^2 + by^2 = 1$ in a finite field

Let $\mathbb{F}$ be a finite field, and let $a,b \in \mathbb{F}$ be given, subject to $a\ne 0, b \ne 0$. Consider the equation $$ax^2 + by^2 = 1.$$ It is guaranteed that there exists a solution to ...
2
votes
0answers
36 views

Showing a Hermitian preserving map is not necessarily positivity preserving?

Say I have a linear map, which is not positivity preserving $$\phi: \mathscr H \to \mathscr H$$where $\mathscr H$ is the set of $n \times n$ Hermitian matrices. Then does there exist a positive ...
5
votes
1answer
120 views

Weighted sum of squares, in a finite field

Let $\mathbb{F}$ be a finite field. Let $a_1,\dots,a_n \in \mathbb{F}$ be given. I want to know whether there exists $x_1,\dots,x_n \in \mathbb{F}$ such that $$a_1 x_1^2 + a_2 x_2^2 + \dots + a_n ...
0
votes
1answer
27 views

Establishing an Upper Bound for a Curious Function

Suppose I have a sequence of positive real numbers $a_1, \, a_2, \, \dots \,a_n$ such that the following is satisfied: $\sum \limits_{i=1}^{n}a_i = 1$ I am trying to find the smallest value of $L$ ...
0
votes
0answers
938 views

Prove that the expectation of residual sum of squares (RSS) is equal to $\sigma^2(n-2)$

The assumed regression model is $E(Y_i|x)=\gamma+\beta(x-\bar{x})$ and $Var(Y|x)=\sigma^2$. So I have: $E(RSS)= E(\sum\limits_{i=1}^n (y_i-\hat{y_i})^2)= E(\sum\limits_{i=1}^n ...
6
votes
2answers
133 views

Are there identities which show that every odd square is the sum of three squares?

I am looking for algebraic identities of the form $$ (2n+1)^2 = f(n)^2 + g(n)^2 + h(n)^2, $$ where the functions are polynomials in $n$. EDIT: Evidently $(6k)^2 = 36k^2$ is trivially the sum of ...
0
votes
1answer
534 views

Dot product of the column vectors from a matrix and their transposes through matrix multiplication

I have a matrix with data, every dataset is a column vector in my matrix. I want to know the dot product of the transpose of each column vector with the original column vector. If I transpose the ...
5
votes
4answers
1k views

Can an integer of the form $4n+3$ written as a sum of two squares?

Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact?
0
votes
5answers
131 views

Relationships between the elements $(a,b,c,d)$ of a solution to $A^2+B^2+4=C^2+D^2$

I have reduced a certain equation (in positive integers) to the equation $$A^2 + B^2 + 4 = C^2 + D^2. \quad(\star)$$ Assume the positive integers $(a,b,c,d)$ are any solution to $(\star)$. Are there ...